Functional Analysis II Phan Thanh Nam Winter 2020-2021 Contents 1 Introduction 4 1.1 Weyl's law and semiclassical estimates ..................... 5 1.2 Laplacian on bounded domains ......................... 6 1.3 Applications to many-body fermionic systems................. 7 2 Basic spectral properties of Schrodinger operators 10 2.1 Hilbert spaces................................... 10 2.2 Self-adjoint operators and Spectral theorem .................. 12 2.3 Min-max principle ................................ 17 2.4 Sobolev inequalities................................ 19 2.5 Schrodinger operators............................... 25 3 Semiclassical estimates 31 3.1 Cwikel-Lieb-Rozenblum inequality........................ 31 3.2 Lieb-Thirring inequality............................. 36 3.3 Birman-Schwinger Principle........................... 42 3.4 Kato-Seiler-Simon and Cwikel's inequalities.................. 48 1 2 CONTENTS 4 Weyl's law 56 4.1 Coherent States.................................. 56 4.2 Weyl's law for sum of eigenvalues........................ 59 5 Dirichlet Laplacian 70 5.1 Berezin-Li-Yau inequality ............................ 72 5.2 Sum of eigenvalues................................ 75 5.3 Distribution of eigenvalues............................ 77 5.4 Polya conjecture ................................. 80 5.5 Weyl's conjecture................................. 85 5.6 Can one hear the shape of a drum?....................... 91 6 Neumann Laplacian 94 6.1 Essential spectrum vs. Compact resolvent ................... 96 6.2 Extension domains................................ 97 6.3 Kröger's inequality................................ 103 6.4 Lieb-Thirring inequality for Neumann Laplacian................ 105 6.5 Weyl's law..................................... 110 6.6 Pölya conjecture ................................. 113 7 Many-body quantum systems 122 7.1 Slater determinants................................ 123 7.2 Reduced density matrices ............................ 126 7.3 Ideal Fermi gas.................................. 133 8 Thomas-Fermi theory 140 8.1 Density functional theory............................. 140 8.2 Convergence of the kinetic density functional.................. 144 8.3 Convergence of the Levy-Lieb functional.................... 151 8.4 Convergence of ground state energy and ground states ............ 156 8.5 Atomic Thomas-Fermi minimizer........................ 160 9 Hartree-Fock theory 167 9.1 Lieb-Oxford inequality.............................. 171 9.2 Error bound for atomic reduced Hartree-Fock energy............. 177 9.3 Bach's correlation inequality........................... 183 CONTENTS 3 9.4 Error bound for atomic Hartree-Fock energy.................. 191 9.5 Hartree-Fock energy of a homogeneous Fermi gas............... 194 10 Correlation energy 202 10.1 Fock space formalism............................... 203 10.2 Particle-hole transformation........................... 210 10.3 Estimates for kinetic and number operators .................. 215 10.4 Removing the non-bosonizable term....................... 220 10.5 Diagonalization of bosonizable term....................... 231 11 Stability of matter 235 11.1 Teller's no-binding theorem ........................... 236 11.2 First proof of the stability of matter....................... 239 11.3 Baxter's electrostatic inequality......................... 241 11.4 Second proof of the stability of matter ..................... 246 11.5 Existence of thermodynamic limit........................ 248 11.6 Grand-canonical stability............................. 252 11.7 Instability for bosons............................... 254 Chapter 1 Introduction In classical mechanics, a particle can be described by a pair of position-momentum (x,p) G Rd x Rd. In quantum mechanics, we cannot determine both position and momentum at the same time (by uncertainty principle). Therefore, a quantum particle has to be described by a normalized function ip G L2(Rd) with |-0(x)|2 = probability density of position, |-0(/c)|2 = probability density of momentum. The semiclassical approximation suggests to relate quantum quantities by classical ones, using the idea that a quantum state a unit volum in the phase space Rd x Rd. For example, for the sum of negative eigenvalues of a Schrodinger operator with a real-valued potential V : Rd —> R we can expect Tr[-A + V(x)]-n J J [|2ttA;|2 + V(x)]_dkdx JWLd JWLd tJ \V(x) ■JWLd l+d/2d; where a_ = min(a, 0) and 4 1.1. WEYL'S LAW AND SEMICLASSICAL ESTIMATES 5 1.1 Weyl's law and semiclassical estimates A cornerstone of the semiclassical analysis for Schrodinger operators is Weyl's law: for all d > 1, if V- G L1+d/2(Rd), then Tr[—A + \V(x)]_ = -Lf4 / \\V(x).\1+d/2dx + o(\1+d/2)x^QC. JWLd By introducing h := A-1/2, Weyl's law is equivalent to Ti[-h2A + V{x)]_ = -h~dLfd [ \V{x)_\1+d'2dx + o{h-d)h^0. This is consistent to Bohr's correspondence principle that the behavior of a system described by quantum mechanics reproduces classical mechanics in the limit h —> 0. In our applications, it is important to have quantitative estimates for finite parameters. In 1975, Lieb and Thirring proved that for all d > 1, if V- G L1+d/2(Rd), then Tr[—A + V{x)]_ > -L14 [ \V{x)_\1+d/2dx. JWLd Here Lld is independent of V. Note that if we just look at the first eigenvalue, then using Sobolev's inequality it is not hard to prove that Ai(-A + V(x)) > -Lf°d f \V(x)_\1+d/2dx. The Lieb-Thirring inequality is deeper! It is related to Pauli's exclusion principle, while Sobolev's inequality is a version of the uncertainty principle. Obviously, the optimal constant in the Lieb-Thirring inequality satisfies Lld > max(L^d, Lf°d). Lieb and Thirring conjectured that , „ (Lfd ifd>3, Lhd = max(Lf4,Lf;d) = { 14 [L^d ifd = l,2. This is an important open problem in spectral theory and mathematical physics. More generally, the semiclassical approximation also applies to the sum of moments of eigenvalues of Schrodinger operators. In particular, the number of negative eigenvalues of 6 chapter 1. introduction —A + Vix) can be approximated by / f t(\27rk\2 + V(x)<0)dkdx = L^d [ \V{x)_f2dx, Lgd = 7^L Weyl's law also extends to this case, namely the approximation holds to the leading order with V XV, X —> +00. For finite parameters, the Cwikel-Lieb-Rozenblum (CLR) inequality states that if d > 3, then the number of negative eigenvalues of —A + V(x) is bounded by L04 [ \V{x)_f2dx. ■JWLd This bound fails in 1 and 2 dimensions. Moreover, in general L0d > Lfd for all dimensions. 1.2 Laplacian on bounded domains Note that in the above discussion, we do not put conditions on V+. In particular, we may consider the hard sphere potential of a bounded domain C Md: 0 if x g n, V{x) = -00 if X ^ £1 The corresponding Dirichlet Laplacian —A has eigenvalues 0 < Ax < A2 < ... with lim^oo A„ = +00. These eigenvalues can be computed using the min-max principle and the quadratic form domain Hq(CI). They also satisfy the equations -Aun = Xnun, un\dQ = 0. Obviously, the number of eigenvalues below A of the Dirichlet Laplacian is the same to the number of negative eigenvalues of —A — A. Therefore, Weyl's law tells us N(X) = L*Xd/2\n\ + o(Xd/2)x^ Lt4= m which is equivalent to A =__n2/d + o(n2/d) The asymptotic formula in this form was first proved by Hermann Weyl in 1911, solving a conjecture of Sommerfeld-Lorentz in 1910. There are also many works for the next order 1.3. APPLICATIONS TO MANY-BODY FERMIONIC SYSTEMS 7 correction (involving |(9Q|), including notable papers of Victor Ivrii in 1980. The eigenvalue problem for the Laplacian on bounded domains was strongly motivated by music, going back to Rayleigh in 1877 with "The Theory of Sound". This has become more popular since Kac's 1966 paper "Can one hear the shape of a drum?". This is an interesting inverse problem: assuming that we know all eigenvalues of the Dirichlet Laplacian on tt, can be determine the shape of tt? Clearly, we can "hear" at least the volume of £1 (or even |<9Q|). But can we hear more, e.g. can we determine £1 uniquely (up to translations and rotations)? An important open problem is Polya's conjecture for the sharp eigenvalue bound A > _^£_n2/d Vn > 1 n ~ {\Bx\\Sl\yiA ' " It is straightforward to see that this bound holds when Q is a cube. Polya proved this bound for tiling domains; however, the problem is still open even when Q is a ball! The best known result in this direction is the Berezin-Li-Yau inequality d (2tt)s J^A„ >Kcl\n\-2'dN1+2l\ Kcl = 71=1 d + 2 iSil2^' This is a consequence of Polya's conjecture, and also a particular confirmation of the Lieb-Thirring conjecture for the hard sphere potential (up to a constant shift). It turns out that the Berezin-Li-Yau inequality can be used to prove a Lieb-Thirring inequality with the semiclassical constant plus some error which is small in applications. 1.3 Applications to many-body fermionic systems From first principles of quantum mechanics, a system of N identical fermions in lRd is described by a (normalized) wave function in L2(M>dN) satisfying the anti-symmetry ty(xi,xl,Xj,xN) = — ^(xi,Xj,xl,xN), Vz 7^ j, Vxj G Rd. A typical many-body Schrodinger operator has the form N HN = ^2(-AXt+V(xl))+ Wixi-xj) i=l KkjXN 8 chapter 1. introduction acting on the anti-symmetric space L^(R ). The ground state energy is en= inf (v,hnv). \\nL2=i If a ground state exists, it satisfies the Schrodinger equation hNty = en^. When n becomes large, this beautiful linear theory is very difficult to compute (even numerically) because there are so many variables. In practice, people often rely on effective theories, which are nonlinear but dependent on less variables. A popular method in computational physics and chemistry is density functional theory, where the complicated wave function \I> : m>dn —> C is replaced by its one-body density p^ : lRd —> [0, oo) p^{x) = n I \^(yx,x2, ...,xN)\2dx2...dxN, / = n. JRd(N-l) JRd The oldest density functional theory is Thomas-Fermi theory (1927), where the ground state energy is computed by et/= inf \kc1 [ p1+2ld+ [ Vp+l [[ p(x)p(y)W(x-y)dxdy J P=N I JWLd JWLd ^ J JWLdxWLd In particular, the Thomas-Fermi approximation for the kinetic energy N r i=i ' j^ can be interpreted as a dual version of Weyl's law for the eigenvalue sum of (one-body) Schrodinger operators. A key concept here is Pauli's exclusion principle which states that we cannot put two fermions in a common quantum state. Mathematically, if we define the one-body density matrix 7^ as a trace class operator on L2(R.d) with kernel jR R we have N r N h, V(-AXJ^\ + / Rp* = h, J2(-AXi + R(xi))*) = Tr((-A + i?)7) X i=i 7 X i=i 7 > V Ai(-A + > Tr[—A + « -L?d / \R.\1+d'2. i=i ' By optimizing over f?, we obtain an appropriate justification for the Thomas-Fermi approximation for the kinetic energy. More generally, under suitable conditions on the interaction potential W, we will show that lim = i. N^oo EN Normally, the Thomas-Fermi theory tells us the leading order behavior of weakly interacting Fermi gas. A better approximation is the Hartree-Fock theory, in which we restrict the wave functions to Slater determinants (ui A u2 A ... A uN)(xi,xN) = ^_ det (ui(xj))i Jif* such that In particular, ||J(x)||^» = Recall that the dual space Jif* contains all linear bounded operators from Jif —> C. Some times, it is convenient to use the bra-ket notation where \x) G //. {x\ = J(x) G 3tf*. 10 2.1. HILBERT SPACES 11 Then (x\y) = (x,y) (the usual inner product) and is a rank-one operator. Definition (Weak convergence). Let {a^K^Li C Jt? and x G Jif. We write xn —^ x, namely xn converges to x weakly in Jif, if (xn,y) ->■ (x,y), My G . Remark: In general, if ,3^ is a normed space, people usually say that xn —^ x weakly in ,3^ if F(xn) ->■ F(x), VF G JT*. When is a Hilbert space, the latter definition is equivalent to the former, thanks to Riesz's representation theorem. In this case, the weak convergence xn —^ x in Jif is also equivalent to the weak-* convergence J(xn) —^* J(x) in Jif*. The concept of weak convergence is very helpful to gain the compactness. / Theorem. Let Jif be a Hilbert space and let {xn}^=1 C Jf. • (Banach-Steinhaus) If {xn} converges weakly in Jif, then {xn} is bounded. • (Banach-Alaoglu) If {xn} is bounded, then there exists a subsequence {xnk}^=1 which converges weakly in Jif. Definition. The Hilbert space Jif is separable if there exists an orthonormal basis {un}n>i (finite or countable). We will always work on separable Hilbert spaces. -\ Theorem. // the Hilbert space Jif is separable and {un}n>i is an orthonormal basis, then we have Parsevel's identity x = 'y^(un, x)un, Vx G Jif. n>l 12 CHAPTER 2. BASIC SPECTRAL PROPERTIES OF SCHRÖDINGER OPERATORS Consequently, \x |2 _ n>l ^2\{un,x)\2, VxG^T. In particular, we have Bessel's inequality: for any orthonormal family {un}, >^2\{un,x)\2, VxG^T. \x |2 Exercise. Let {un}^=1 be an orthonormal family in a Hilbert space . Prove un —^ 0 weakly in . 2.2 Self-adjoint operators and Spectral theorem Definition (Unbounded operators). An operator A on ,3^ is a linear map A : D(A) —> J{? with a dense, subspace D(A) C Jt? (domain of A). The adjoint operator A* : D(A*) ->■ 3tf is defined by D(A*) = |x G 3tf \3a*x G 3tf : {x,Ay) = {A*x,y), Vy G £>(A)}. The operator A is self-adjoint if A = A*. Definition (Spectrum). Let A : D(A) —> Jrf? be an operator on a Hilbert space Jff. Its spectrum is cr(A) = C\{A G C : (i - ^)_1 *s a bounded operator}. We can decompose a(A) = adis(A) U aess(A), adis(A) n aess(A) = 0, where the discrete spectrum adis(A) is the set of isolated eigenvalues with finite multiplicities and the essential spectrum uess(^4) is the complement. Note that the spectrum and the essential spectrum are always closed sets. 2.2. SELF-ADJOINT OPERATORS AND SPECTRAL THEOREM 13 Exercise. Let £1 C Rd be a Borel set, /i a locally finite Borel measure on Cl, and a g Lfoc(Tl, fi) a real-valued function. Consider the multiplication operator Ma on L2(£l,/i) defined by (MJ)(x) = a(x)f(x), D(Ma) = {/ 6 L2(£l, /i), af g L2(Q^)}. Prove that (i) Ma is a self-adjoint operator and a(Ma) = ess-range(a) C R, namely Ag 0, Ve > 0. (ii) A is an eigenvalue of Ma iff yu(a_1(A)) > 0. (iii) A g Odis(^a) iff A is an isolated point of a(Ma) and 0 < yu(a_1(A)) < oo. A cornerstone of spectral theory is the Spectral theorem which says that any self-adjoint operator is unitarily equivalent to a multiplication operator. Theorem (Spectral theorem). Assume that A is a self-adjoint operator on a Hilbert space J4f. Then there exists a Borel set C M.d for some d > 1, a locally finite Borel measure /i on Cl, a real-valued function a G L^c(£l,/i) and a unitary transformation U : 3tf ->■ L2(tt,/i) such that UAU* = Ma. Here Ma is the multiplication operator on L2(tt,[i), defined by (Maf)(x) = a(x)f(x), D(Ma) = {/ G L2(Q, fi),af G L2(Q, /i)}. We can choose £1 = cr(A) X N, a(\,n) = A and /i being a locally finite measure. In this case, A is an eigenvalue of A if and only if there exists n G N such that /i({(\,n)}) > 0; moreover, the number of such n's is equal to the multiplicity of A. Remark: As a consequence of the spectral theorem, if A is self-adjoint, then cr(A) C 1R. Given a self-adjoint operator A on a Hilbert space J^f and a smooth function / : R —> R, we 14 CHAPTER 2. BASIC SPECTRAL PROPERTIES OF SCHRÖDINGER OPERATORS can define the self-adjoint operator f(A) on Jif by Uf(A)U* = fiUAU*) = f(Ma) where U : Jt? —> L2(tt) is a unitary transformation making UAU* = Ma. This is called the functional calculus. Moreover, the spectral theorem can be used to prove several abstract results for self-adjoint operators. / Exercise. Let A : D(A) —> Jff be a self-adjoint operator on a Hilbert space Jff. Prove that A is a bounded operator if and only if D(A) = Jff. ) - \ Exercise. Let A : D(A) —> Jrf? be a self-adjoint operator on a Hilbert space J$?. Prove that the following statements are equivalent: (i) A > 0, namely {u, Au) > 0 for all u 6 D(A); (i) a (A) C [0,oo). -> Exercise (Weyl's Criterion). For any self-adjoint operator A on a Hilbert space Jff, prove that the following statements hold true: (i) A G o-{A) iff there exists a Weyl sequence {un} C D(A) such that \\un\\ = 1, 11(^4 — A)-u„|| —^0 as n —> oo. (ii) A G aess(A) iff there exists a Weyl sequence {un} C D(A) such that {un} an orthonormal family, \\(A — A)-u„|| —^0 as n —> oo. Definition. Let A be a bounded operator on a Hilbert space Jff. We say that A is a compact operator if A maps any bounded set to a pre-compact set. ^l£xercise^Le^Al)e(^)mn^ 2.2. SELF-ADJOINT OPERATORS AND SPECTRAL THEOREM 15 operator iff A maps weak convergence to strong convergence, namely (xn x weakly in ,3^) ==> (Axn —> Ax strongly in Jff). Theorem (Spectral theorem for compact operators). Let A be a self-adjoint compact operator on a separable Hilbert space Jff. Then A has an orthonormal eigenbasis {un} C Jrf? with eigenvalues {\n} C R and \n —> 0. In short, we have the spectral decomposition A = y^\n\un){un\. n>l Exercise. Let {un}^=1 be an orthonormal family in a Hilbert space Jff. Let {\n}^Li C R be a bounded sequence. Consider the operator A : = ^ \n\un){un\. n>l (i) Prove that A is a bounded, self-adjoint operator. (ii) Prove that A is a compact operator if and only i/lim^oo A„ = 0. In practice, the self-adjointness is not always easy to verify. A weaker concept is Definition. An Operator A on a Hilbert space Jrf? is Symmetrie if {x,Ay) = {Ax,y), Vx, y G D(A). This is also equivalent to {x,Ax)eR, xeD(A). Obviously, if A is self-adjoint, then A is symmetric. But the reverse is not true. Two useful methods to find self-adjoint extensions for symmetric operators are Kato-Rellich theorem and Friedrichs' extension. ^Theorern^^Z^o^Rfilh^hth 16 CHAPTER 2. BASIC SPECTRAL PROPERTIES OF SCHRÖDINGER OPERATORS Jif. Let B be a symmetric operator on Jif satisfying \\Bx\\ < (1 - + Cs\\x\\, Vx G D(A) C D(B), for some constant e > 0 independent of x (we say that B is A-relatively bounded with the relative bound 1 — e). Then A + B is a self-adjoint operator on ,3^ with domain D(A + B) = D(A). Exercise. Let A be a self-adjoint operator on a Hilbert space. Let B be a symmetric operator which is A-relatively compact, namely D(A) C D(B) and B(A + is a compact operator. Then A + B is self-adjoint on D(A) and aess(A + B) = aess(A). Hint: You can write B = B(A + i)_1(A + i) and use Weyl's Criterion. Theorem (Friedrichs' extension). Assume that A is bounded from below, namely {x,Ax) >-C\\x\\2, Vx£D(A) with a finite constant C independent of x. Then A has a self-adjoint extension Ap which has the same quadratic form domain Q(AF) = Q(A). Recall that we define Q(A) as the closure of D(A) under the quadratic form norm \\x\\q(a) = \/{x,(A + C + l)x). Note that in general the domain D(Ap) is often not known explicitly (unlike the extension given by Kato-Rellich theorem). Nevertheless, all eigenvalues below the essential spectrum of Ap can be computed without knowing the domain D(Ap), thanks to the min-max principle. In particular, the Friedrichs extension preserves the ground state energy inf {x,Ax) = inf (x,A0x). x€D(A),\\x\\=l x£D(A0),\\x\\=l In fact, the Friedrichs extension is the largest possible extension of an operator (in the sense Krein's characterization). 2.3. MIN-MAX PRINCIPLE 2.3 Min-max principle 17 Theorem (Min-Max Principle). Let A be a self-adjoint operator on a Hilbert space . Assume that A is bounded from below and define the min-max values /in(A) = inf sup {u,Au). MdD(A) „eM dim M=n 11 ti 11 =1 Then /in(A) is an increasing sequence and its the limit fi^A) := lim^-^ iin(A) < +oo satisfies Hoo(A) = inf aess(A). Moreover, if /in(A) < fi^A), then /ii,.. .,/in are the lowest eigenvalues of A. Remarks: • Here we use the convention that if uess(^4) = 0, then inf<7ess(^4) = +oo. • In the above definition, the condition M C D(A) can be replaced by M C S for any subspace 2) which is dense in the quadratic form domain Q(A). Thus if A is the Friedrichs' extension of a (densely defined) operator A0, then the min-max values can be computed using the domain D(A0). • The min-max values is monotone increasing in operator, namely if A < B, then Hn(A) < tin(B), Vn = l,2,... Proof. Step 1. We prove that fi^A) < inf aess(A). We take A G aess(A) and prove that fJ-n(A) < A, Vn > 1. By Weyl's criterion, since A G aess(A), there exists an orthonormal family {nm}™=1 C D(A) such that \\(A — \)um\\ —> 0. Consider the space Mm:n = Span(-um+i, ...,um+n), dimMm>n = n. 18 CHAPTER 2. BASIC SPECTRAL PROPERTIES OF SCHRÖDINGER OPERATORS By the definition of the min-max values, /J-n(A) < sup {u,Au). ||u|| = l On the other hand, since linv^^ \\(A — A)-um+fc|| = 0 for all k = 1,2, ...,n , we have lim sup (u, Au) = A. ||u||=l Therefore, /in(A) < A. This holds for all n > 1, implying that /ioo(A) < inf aess(A). Step 2. We prove that if /i\(A) < /100(A), then /ii(A) is the lowest eigenvalue. Using ^00(^4) < inf o"ess(A) from Step 1, we find that /ii(A) ^ eress(^4). On the other hand, A*i (A) = inf (u,Au) = mi a (A) £ a (A). \\u\\=l Thus Hi(A) £ adis(A), namely it is an eigenvalue with finite multiplicity. Clearly it is the lowest eigenvalue. Step 3. We prove that if /12(A) < /100(A), then /12(A) is the second lowest eigenvalue. By Step 2, we know that a*i(-<4) is an eigenvalue with an eigenvector u±. Then A leaves invariant the space J^i = {wi}"1 and we can define A\ = A\^ as as an operator on Note that thanks to the decomposition A = AJi (A) K) (iii I ®A-i we find that a (A) = {/ii(A)} U a(At) and /ii(Ax) =/i2(A) (why?). Consequently, /12(A) £ cr(Ai) C cr(A). Thus the condition /12(A) < /ioo(A) and the inequality /ioo(A) < inf<7ess(^4) from Step 1 imply that /12(A) £ 0dis(-<4)) namely it is an eigenvalue of A. Moreover, /12(A) = fii(Ai) the lowest eigenvalue of A\, and hence /12(A) is the second lowest eigenvalue of A. Step 4. By the same argument, we can prove that if /in(A) < /ioo(A), then /in(A) is the n-th lowest eigenvalue of A. Moreover, if /in(A) < /in+i(A) = /ioo(A), then /ioo(A) £ cr(A). Thus in all cases, all min-max values /in(A) belong to —^ii(A). In this case we say that A has compact resolvent. As a consequence, the eigenfunctions of A form an orthonormal basis. Exercise. Let A be a self-adjoint operator on a Hilbert space Jif. Assume that A is bounded from below and let /in(A) be its min-max values. Prove that for all N £ N, N N Hn{A) = inf I 'y^(un, Aun) : {un}^=1 an orthonormal family in J^j. n=l n=l 2.4 Sobolev inequalities Next, we turn to the fact that the Schrodinger operators are defined on the real space m>dn. Therefore, we recall some standard results from real analysis. Definition (Sobolev Spaces). For any dimension d £ N and s > 0 (not necessarily an integer), define Hs(Rd) := {/ £ L2(Rd) | \k\"f(k) £ L2(Rd)} with f the Fourier transform of f. This is a Hilbert space with the inner product {f,9)H- = / f(k)g(k)(l + \2irk\2")dk Remarks: 20 CHAPTER 2. BASIC SPECTRAL PROPERTIES OF SCHRODINGER OPERATORS • We use the following convention of the Fourier transform f(k) = [ < Jm.d In this "engineering convention" we have the inverse formula f(x) = [ e2mk-*f(k)dk and the the Plancherel theorem ||/||L2(Rd) = ||/||L2(Rd)- • On the Sobolev space Hs(M>d), the weak derivative is defined via the Fourier transform D°~f(h) = (2mk)af(k) E L2(Rd) for any multiple index a = (a±,a^) with \a\ = a\ + ... + < s. • In the course we will mostly think of s as an integer for simplicity. The non-integer case (the so-called fractional Sobolev spaces) is useful for studying relativistic quantum mechanics. Theorem (Sobolev Inequalities/Continuous > embedding). Let d > 1 and s > 0. Then 11/ LP(Kd) < C||/ ffS(n ld), V/GF(Kd) where 2 < n < 2d ifs < d/2, < 2 ^ p < oo, ifs = d/2 2 ^ p ^ oo, ifs> d/2. We say that Hs(rd) C LP(M d) with continuous embedding. When s > d/2 we also have the continuous embedding Hs(rd) C ^(w1) (the space of continuous functions with sup-norm). Remarks: 2.4. SOBOLEV INEQUALITIES 21 • In the case s < d/2, the power p* := 2d/(d — 2s) is called the Sobolev critical exponent. In fact, this is the only power works for the following standard Sobolev inequality LPw) 0, the Sobolev inequality becomes weaker in higher dimensions. For example, h\r) c l2(r) n^(M), h1^2) c P| lp(r2), h1^3) c p| lp(r2). 2 E)\u(k)\2 Jm? Jo r dE / dA;l(|27rA;|2s > E)\u{k)\2 Jm? o dE / dx\uE+(x)\2 = dx dE\uE+(x)\2 o Jm? Jm? Jo where the function uE+ is defined via the Fourier transform uE+{k) = 1(|2ttA;|2s > E)u{k). When d > 2s, we have the uniform bound \u(x) - uE+(x)\ = [ dkel2nk-xu(k)l(\27rk\2s < E) 22 CHAPTER 2. BASIC SPECTRAL PROPERTIES OF SCHRODINGER OPERATORS < 2irk\2s\u(k)\' 1/2 dkl(\2irk\2a < E)\2irk\ -2s 1/2 „ 1 d-2s = CnK?E— with a constant Co depending only on d and s. By the triangle inequality, \-n,E+(^ \uE+(x)\ > \u(x)\ — \u(x)—uE+(x) > \u(x)\ — \ \ / 1 1 > uixM-CoK^E^1 Of course this bound is nontrivial only when E < \u(x) \ \ d=k. C0K2 Integrating over E 6 (0, oo) we get dE\uE+(x)\2 > / dE o Jo = CAuix) 00 2 \u(x)\ — CnK^E^'1 \u(x)\\1^2. K-2 2d 2a = Ci\u(x)\d-2° K d-2s. In conclusion, which is equivalent to K > dE'—* Kä=2S > Ci i r \ i 2d \u(x) d"2s \U(X)\d-2s . Inserting the definition K = ||(—A)s/2f\\22(„d) we arrive at the desired inequality. □ Theorem (Sobolev compact embedding). Let d > 1 and s > 0. Then for any bounded set Q C Rd, the operator tQ : Hs(Rd) ->■ Lp(lRd) is a compact operator, where 2d 2 d/2. When s > d/2, we also have the compact embedding 1q : H sfTO>d\ Remark: The Sobolev compact embedding means that if un —^ u weakly in Hs{Rd) with 2.4. SOBOLEV INEQUALITIES 23 s > 0, then for any bounded set flcR11, Uun -)■ lnu strongly in Lp(Rd). They key point is the strong convergence in L2; and the convergence in LP follows by a standard interpolation (for which we have to avoid the end-point). This kind of result can be interpreted as the operator 1q(1 — A)_s/2 is compact on L2(IR2). We have the following more general result. t \ Theorem. Let /, g G L°°(lRd) and f(x) —> 0 and g(x) —> 0 as \x\ —> oo. T/ien f(x)g(—i'V) is a compact operator on L2(M.d). Here f(x) is the usual multiplication operator and g(—iS7) is defined by the spectral theorem, or equivalently via the Fourier transform: This theorem can be interpreted in the same spirit of the uncertainty principle: localizing both position and momentum gives us a compact operator. Further estimates for the operator f(x)g(—iV) will be discussed in the next chapter. Proof. We prove that if un —^ 0 weakly in L2(lRd), then vn{x) = f{x){g{-iS/)un){x) ->■ 0 strongly in L2(Rd). Step 1. Let us consider the case when / and g are compactly supported. We write JWLd Since un —^ 0 weakly in L2(lRd), un —^ 0 weakly in L2(lRd). Since g is bounded and compactly supported, el27rfe'x#(27r/c) G L2(Rd,dk). Thus (g(-iV)un)(x) ->■ 0, for a.e. x G lRd. Moreover, by Holder inequality we also know that g(—iV)un is bounded in L°°(lRd). Since / is bounded and compactly supported, we find that (g{-iV)u){k) = g{2irk)u{k). vn{x) = f{x){g{—iS/)un){x) —> 0, for a.e. x G Rd 24 CHAPTER 2. BASIC SPECTRAL PROPERTIES OF SCHRODINGER OPERATORS and IKIlL°°(Rd) < C, SUpp(u„) C SUpp/. Thus vn —> 0 strongly in L2(lRd) by Lebesgue Dominated Convergence Theorem. Step 2. Now we consider the case when g is compactly supported and fix) —> 0 as |x| —> oo. Then for any e > 0, we split f = fs + fs with f£ being compactly supported and ||/£||l°° < £• By the triangle inequality \\f(x)g(-iV)un\\L2 < \\f£{x)g{-iV)un\\L2 + \\f£g{-iV)un\\L2 < \\f£(x)g(-iV)un\\L2 + II/eIIl-IMUHWUs < \\fe(x)g(-iV)un\\L2+Ce for a constant C independent of e and n. Here we used the fact that un is bounded in L2 since it converges weakly in L2. By Step 1, lim \\fs{x)g{-iV)un\\L2 = 0. Thus lim sup ||/(x)^(-iV)ii„||L2 < limsup ||/e(x)^(-iV)ii„||L2 + Ce = Ce Since this holds for any e > 0, we conclude that iV)un\\L2 —> 0. Step 3. Now we consider the case when fix), g{x) —> 0 as |x| —> oo. Then for any e > 0, we split 9 = 9e + 9e with g£ being compactly supported and | 11 5: £• By triangle inequality \\f{x)g{-iS/)un\\L2 < \\f{x)g£{-iS/)un\\L2 + ||/(z)<7e(-iV)ii„||L2 < ||/(^)^(-*V)li„||L2 + ||/||l~||<7£||l°°|K||L2 < \\f(x)g£(-iV)un\\L2+Ce. Taking n —> oo, and then e —> 0, we find that \\f(x)g(—iSI)un\\L2 —)■ 0. □ 2.5. SCHRODINGER OPERATORS 25 Exercise. Let f G L°°(Rd) such that f(x) ->■ 0 as \x\ —> oo. Prove that if un —^ 0 weakly in the Sobolev space Hs(Rd) for some constant s > 0, i/ien /itn —>■ 0 strongly in L2(Rd). f Exercise. Let F, G : Rd —>■ R 6e locally bounded functions satisfying F(x),G(x) —> oo as \x\ —> oo. Prove that the operator F(x) + G(—iS7) on L2(R.d) has compact resolvent. 2.5 Schrodinger operators Now we are ready to discuss some basic spectral properties of the Schrodinger operator —A + V(x) on L2(Rd). We will always assume that V is a real-valued potential. First, consider the case when V is bounded or vanishing at infinity. Theorem. Assume that V G Lp(Rd) + Lq(Rd) with oo > p,q > max(2, d/2), then —A + V is a self adjoint operator on L2(Rd) with domain H2(Rd) and aess(-A + V) = [0,oo). More generally, the self-adjointness still holds if oo > p, q > max(2, d/2) when d ^ 4 and oo > p, q > 2 when d = 4; and the essential spectrum property still holds if oo > p > 2 when d < 3 and oo > p > d/2 when d > 4. Remark: Ls(Rd) C Lsl(Rd) + LS2(Rd) if Sl < s < s2. Proof. Step 1. First we prove the self-adjointness. We use the Kato-Rellich theorem and show that V is A-relatively bounded, more precisely \\Vu\\L2 < e||Aii||L2 +Ce\\u\\L2, Ve > 0, Vit G L2(Rd). We can always write V = V1 + V2, HKlliP 0, V-u G L2(Rd). This is equivalent to the desired estimate (we can change Ce i—> e). Case d > 4: Using the Sobolev embedding H2(Rd) C L~(JRd) and Holder's inequality we can bound \\vm?i? = J|Vi|2H2^ (/l^l2r)7 (/ l«l2r')r' p>2ifd<3 and oo > p > d/2 if d > 4. Take un —^ 0 weakly in L2, then we have to prove that V{\ — A)-1-^ —> 0 strongly in L2. 2.5. SCHRODINGER OPERATORS 27 Note that un —^ 0 weakly in L2 implies that (why?) /„ := (1 - A)"V 0 weakly in H2(Rd). By Sobolev compact embedding, we know that for any bounded set Cl C Rd, then W„ ->• 0 strongly in Ls(Rd) for any s < oo if d < 3 and s < 2d/(d — 4) if d > 4. In particular, given the condition oo>j9>2ifd<3 and oo > p > d/2 if d > 4, we can choose s such that 2 2 - + - = 1. p s Moreover, we know that fn is bounded in Ls(Rd) due to Sobolev continuous embedding. Let us split V = Vi + V2, V! = V1{\X^R}, V2 = V1{\X\>R}. Then 11^112= f f \V\2\fn\2)1/2 < ( [\V\pYP ( [ \fn\sY^0 asn^oo \x\^R and |x|<Ä \\v2fn\\2<[ \v\py[ \fn\8) 0 strongly in L2(Rd) when n —> oo. □ The condition on V can be relaxed slightly if we use Friedrich's extension. Exercise. Assume that V+ G L\oc(Rd) and V- G Lp(Rd) + Lq(Rd) with oo > p, q > max(l, d/2) when d ^ 2 and oo > p, q > 1 when d = 2. Prove that: • For all u G H^(Rd) and e > 0, \V\\u\2 < e I \Vu\2 + CF I \u\2. 28 CHAPTER 2. BASIC SPECTRAL PROPERTIES OF SCHRÖDINGER OPERATORS • The operator — A + V with core domain C%°(M.d) is bounded from below on L2(Rd). Hence, it can be extended to be a self-adjoint operator by Friedrichs' method. • If we assume further that V G Lp(M.d) + Lq(M.d) with oo > p, q > max(l, d/2) when d 7^ 2, and oo > p, q > 1 when d = 2, then the quadratic form domain of —A + V is H1^) and aess(-A + V) = [0,oo). Hint: For the essential spectrum, you can use Weyl's criterion. In particular for d = 3, the Friedrich extension covers the case V(x) = — \x\~s with 0 < s < 2, while the Kato-Rellich theorem requires 0 < s < 3/2. In the critical case s = 2, we have Hardy's inequality 2 f \Vu(x)\2dx > - f ^j^dx, VueH Jm? 4 JM3 \x\ Next, we show that if the potential V(x) is negative and decays slowly, then —A + V has infinitely many negative eigenvalues. The opposite regime, when —A + V has finitely many eigenvalues, will be studied in the next chapter. Theorem. Assume that V G LP(Rd) + Lq(Rd) with oo > p, q > max(l, d/2) and V(x) ^ — \x\~s, 0 < s < 2, for \x\ large. Then — A + V has infinitely many negative eigenvalues. Here — A + V is the self-adjoint operator obtained by Friedrichs' extension. Proof. By the min-max principle, it suffices to show that all the min-max values are negative: /in := inf max {u, (—A + V)u) < 0. dimM=n mGM IMIL2=1 To choose M, we take normalized functions u% G + 1 > \x\ > i), i = 1, 2, ...,n. Then take R > 0 and define M := Span(u\R\i = l,2,...,n), u\R)(x) := R'^u^x/R). 2.5. SCHRODINGER OPERATORS 29 Clearly dimM = n since {u\ }™=1 is an orthonormal family. Indeed, they have disjoint supports and in their supports (n + l)R > \x\ > R. Using Vix) ^ — \x\~s with 0 < s < 2 for |x| large, we find that for R large, (u<«>, (_A + V>f>) = J\Vu\R)\2+ f V\uf\2< J |VU'fi,|2- f \x\-\u\R\x)\2te Next, since {w^KLi have disjoint support, we find that max (u, (-A + V)u) = max (uf \ (-A + V)u\R)) < 0. uGM i=l,...,n IMIl2=i This completes the proof. □ Next, we consider the case when the potential Vix) grows to oo at infinity. Theorem. Assume that V E L^oc(Rd) with p > max(l,d/2) when d ^ 2 and p > 1 when d = 1; moreover V(x) —> oo as \x\ —> oo. Then the operator —A + V with core domain C£°(M.d) is bounded from below and can be extended to be a self-adjoint operator by Friedrich's extension. This self-adjoint operator has compact resolvent. Proof. The condition V G Lfoc(lRd) and V(x) —> oo implies that for the negative part, V- G Lp(Rd). Therefore, —A + V > i(-A) + y+-C. Thus —A + V is bounded from below and hence it can be extended to be a self-adjoint operator by Friedrich's extension. It remains to prove that — A + V has compact resolvent. By the min-max principle, it suffices to show that —A + V+ has compact resolvent. We prove by contradiction: assume that a finite element A G cress(—A + V+). Then by Weyl's criterion, there exists an orthonormal family {un}^=1 in L2(Rd) such that ||(—A + V+ — A)-u„||L2 —)■ 0. Consequently, ViiJ2 + / VAuJ2 ->■ A. 30 CHAPTER 2. BASIC SPECTRAL PROPERTIES OF SCHRODINGER OPERATORS Since un is bounded in i/1(IRd) and un —^ 0 in L2(Rd), we have un —^ 0 in i/1(IRd) (why?). Hence, by Sobolev's embedding theorem, for any R > 0 we have Consequently, Therefore, lim ||l{|x|<Ä}^n||L2 = 0. lim ||l{|x|>Ä}^n||L2 = 1- A > lim / V+Iitnl > lim inf 1/(y) |y|>Ä ^ |-u„(x)|2dx = inf V(y). Then sending R —> oo we obtain A = oo, which is a contradiction. □ Chapter 3 Semiclassical estimates 3.1 Cwikel-Lieb-Rozenblum inequality By semiclassical approximation, the number of negative eigenvalues of the Schrodinger operator —A + V(x) on L2(lRd) can be related to its phase-space analogue [ [ t(\27rk\2+ V(x) <0)dkdx = -^- [ \V(x)_\d/2dx. Recall t_ = min(t, 0). The following bound justifies this relation as a universal upper bound. f d \ Theorem (Cwikel-Lieb-Rozenblum (CLR) inequality). If d ^ 3 andV_ G L2(Rd), then M(-A + V)^Cd J |V1|1 Here ftf(—A + V) is the number of negative eigenvalues of —A + V{x) on L2(M.d). The constant d is finite and independent ofV. Remarks: • Here the condition VI G L^(M.d) ensures that —A + V is bounded from below, and hence it can be extended to be a self-adjoint operator on L2(lRd) by Friedrichs' method. • It is not surprising that the positive part of V does not appear in the upper bound because A/*(—A + V) < J\f(—A + VI) by the min-max principle. In general, we do not any serious condition on V+, for example V+ G L11oc(lRd) is sufficient for the operator to 31 32 CHAPTER 3. SEMICLASSICAL ESTIMATES be well-defined in the core domain C%°(M.d). In relation to the Laplacian on bounded domains (we will discuss later), it would be also useful to think of a hard sphere potential where V = +00 outside G Rd (in this case the underlying Hilbert space will be L2(Q)). The following proof is due to Frank, based on Rumin method. This is an extension of the previous proof of Sobolev's inequality to orthogonal functions. Proof. Assume A/* (—A + V) ^ N. Let W be the space spanned by eigenfunctions of negative eigenvalues of —A + V. Since dim W ^ N and \/—A has a trivial kernel, we get dim(y/-~AW) ^ N. Thus we can choose N orthonormal functions in -\/—AW, says \/—Aut. Thus {ui}f=1 C W, {ui, -Auj) = {V-Aui, V-Auj) = öij. Therefore, 0 > J2{uu (-A + V)ui) =N+ fvP>N- [\V-\p, p{x) := J2 i=i J •> i=i For any E > 0, we introduce the function u% + via the Fourier transform max uf+(k) = 1(|2ttä;|2 > E)ui(k). Then similarly to the proof of Sobolev's inequality, we can write n „ n °? „ „ °? n N = ^2 \VUl\2dx = '£2 dE dx\uf+(x)\2 = dx dE22K+(x^2 i=l. i=l i=l By the triangle inequality for the Euclidean norm in Cd, we have n £i«f+(*)i2> 1=1 n E i=i uax n i=l \Ui(x)-uf+(x^2 3.1. CWIKEL-LIEB-ROZENBL UM INEQ UALITY 33 and hence N £l«f+(*)l2> i=i N \ Si"1 \ 1=1 uax) — uf+{x)\2 On the other hand, for d > 3 we have the uniform bound N uf+(x)\2 = 1=1 N E %=i N E dkui(k)t{\2wk\HEy 2nikx i=l dk^irk^ik)- 1 {\2irk\^E} Onikx 1 {|27Tfc|2<£} \2irk\2 \2irk\ dk < CdE~. Bessel Here we have used Bessel's inequality and the fact that {\27rk\ul(k)}iL1 is an orthonormal family in L2(M>d,dk) (as {V—A-ul}^1 is an orthonormal family). Thus AT i=l a/pR - cáE- 2 1 dEdx > — I p(x)d~2. + w Therefore we conclude that ^< / \V.\p<\\V.\\Ld/4p\\L_^_ < \\V-\\Ld/2(CdN which implies N < 11VE id/2 \Ld/2- The following exercise shows that the CLR bound fails if d < 2. □ Exercise. Let d = 1,2. Let V G Lx(lRd) if d = 1 and V G Lx(lRd) n Lp(IRd) /or some jo > 1 if d = 2. Prove that if ' V < 0 T/iera —A + V has at least one negative eigenvalue. Hint: You may consider u£{x) g-^M when d = 1, and u£{x) = e~^1+][X^E when d = 2. Nevertheless, we have a modified result for d < 2. Recall that for any function F : [0, oo) [0, oo] we define the Legendre transform F* : [0, oo) —> [0, oo] by F*(x) = sup{xy-F(y)}, Vx > 0 y>o 34 CHAPTER 3. SEMICLASSICAL ESTIMATES Note that F > G, then F* < G*. Moreover, if F(x) = xp/p then F*(x) = xq/q with 1/p + 1/q = 1, thanks to Young's inequality. Theorem (Bound states in one and two dimensions). For d = 1, 2 and any L > 0, M(-A + V + L)< f dxF*(2lV^}-l)Ld/2. JWLd V L J where A/* (—A + V + L) is i/ie number of negative eigenvalues of —A + V + L on L2(Rd) and Ve^-l), i/d = 2, £tan2(£) + ool(£ > tt/2), if d = 1. F(t) Proof. We proceed similarly to the proof of the CLR bound, except —A will be replaced by —A + L. More precisely, assume A/*(—A + V + V) > N. Let W be the space spanned by eigenfunctions of negative eigenvalues of —A + V + L. Since dim W ^ N, we have dim(\/—A + LW) ^ iV. Thus we can choose N functions {ui\f=1 such that Therefore, AT TV 0 ^ (~A + ^ + LH) = N+ Vp>N- \V-\p, p(x) := i=i J •> i=i For any E > 0, we introduce uf+(k) = l(|27r/c|2 + L > E)ut(k). Then AT „ °? TV £+^)|2 i=l > J dx J dE 0 0 AT n,(x) -n£+(x)l2 since {nl(A;)v/|27rA;|2 + L}l7I 1 is an orthonormal family, N E i=i -u,(x) — uf+(x)\2 = N E %=i d/cnl(/c)l{|27rfc|2+L<£;} 3.1. CWIKEL-LIEB-ROZENBL UM INEQ UALITY N E i=i A;VI ' y/\2irh\* + L Be^sel f l{|27rfc|2+L<£} - |2tt£;|2 + L C When d = 2: it is straightforward to see that when E > L, It L{|27rfc|2+L<£} |2ttA;|2 + L 2tt 1 /-^^ r 6r = —\og(e/l). r2 + L 47r Thus iV > / dx j dE l 47T \og{E/L) = L JdxF(p(x)) with F(t) = £(e47rt - 1). Therefore 0>2iV-2/ \V-\p>N + L dxF(p(x))-2 dx\V_(x)\p(x) = N + l J^dx (f(p) - ^\V_\p^j > N -L F* 2\V_ Thus N < L F* 2\V_ When d = l: if E > L, E Mi2^w}dk = i r E~L i = i arctan , , ~ _ |2ttA:|2 + L 7t70 r2 + L n^L VV L Thus iV > I dx J dE l ' 1 / /£ --=arctan a/--1 tt\/Z VV L > L / dxp(x) tan2(\/Zp(x)) = \/Z /" dxF(V~Lp(x)) 36 CHAPTER 3. SEMICLASSICAL ESTIMATES where F(t) = £tan2(£) if t < tt/2 and oo if t > tt/2. Therefore 0>2iV-2 / \V-\p>N + ^/L / 6xF(y/Zp(x))-2 / dx\v-(x)\p(x) R = N + VT J dx ^F(vTp) - ^^v^Lp^ L Thus N 1 and s>l/2, ifd=l s > 0, ifd = 2, s > 0, ifd>3. If the operator —A + V{x) on L2(M.d) has negative eigenvalues p,\ < p2 < then \V- S+2. n>l " Here the constant Ls^ is finite and independent ofV. The range of s is optimal. Remark: This inequality was first derived by Lieb-Thirring in 1975 for s = 1 in their proof of the stability of matter. Then they extended the inequality to the cases s > 0 when d>2 and s > 1/2 when d = 1. The case s = 0 when d > 3 was proved independently 3.2. LIEB-THIRRING INEQUALITY 37 by Cwikel, Lieb, Rozenblum (CLR bound) around 1977. The case s = 1/2 when d = 1 was solved by Weidl in 1996. Assuming that —A + V can be defined as a self-adjoint operator, it is convenient to write the sum of moments of eigenvalues as ^K(-A + V0|s = Tr|(-A + V0_ n>l where the negative part (—A + V)_ is defined by the spectral projection. Similarly to the CLR bound, in the Lieb-Thirring inequality only the negative part VI is relevant. This follows from the fact that —A + V > —A + VI, leading to Tr | (—A + V^)_|s < Tr |(—A + VL)_|S by the min-max principle. This bound agrees to the semiclassical approximation Tr |(-A +1/)_|s ~ [ [ \{\2irk\2+ V{x))_\sdkdx = Lf4 [ |Vl|s+d/2dx. Jwtd JWLd ' Jwtd where d = 1 T(s + 1) ^s,d , \d -nf n i 1 i d\' (47r)f r(s + i + f) When s > 3/2, the best constant in the Lieb-Thirring inequality coincides to the classical constant Ls4 = Lfd. This was proved by Lieb-Thirring (1976) for d = 1 and extended by Laptev-Weidl (2000) to all d > 1. • It is known that if s < 1, then Ls4 > Lfd. When d = 1 and s = 1/2, Hundertmark-Lieb-Thomas (1998) proved that the sharp constant is £1/2,1 = 2L^2 v • In the most interesting case s = 1, it is conjectured that L\4 = Lfd for d > 3 (we will come back to this case). When d < 2, the conjectured value of Ls4 is given by a one-bound-state/Sobolev inequality. Currently, the best known result is L\4 < lA56Lfd for all d > 1; see FHJN (2018). Proof. We will use the bound on the number of negative eigenvalues derived in the previous section. The proof covers all cases except the critical case d = 1 and s = 1/2. Our starting point is the layer-cake representation: for any s > 0 Tr |(-A + V)_\s = s / A/"(-A + V + E)Es~1dE Jo 38 CHAPTER 3. SEMICLASSICAL ESTIMATES where J\f(—A + V + E) is the number of negative eigenvalue of —A + V + E (which is the same to the number of eigenvalues < — E of —A + V). This formula is analogous to / \f(x)\aMx) = * r»({x ■ 1/0*01 > EjjE^dE. JQ JO Case d > 3: Inserting the CLR bound JV(-A + V + E) < Cd [ \{V{x) + E)_\Ux JWLd to the above layer-cake representation and using Fubini's theorem we obtain for all s > 0 Tr |(-A + V-)-\a =s J\f(-A + V + E)Es~1dE Jo 7r/2), if d = 1. (e/2) ; v 2 Hence, Tr |(-A + Vl)_|s = s / A/"(-A + y + E)Es~1dE Jo = ca,df dx\V{x).\s+í Í dyF* (^-2]ys-1+l Jri Jo \y J 3.2. LIEB-THIRRING INEQUALITY 39 In the last equality we have changed the variable E = 2\V(x)_\y. It remains to show that dyF* | - - 2 | ys~1+^ < oo. o \y J Consider the case d = 2, s > 0. Recall that if F > G, then F* < G*. Using oo F(t) = t(e4nt - 1) > Cptp, V2 < p < we find that F*(t) < Cqtq, \/l 0, we can take 1 < q < 1 + s (such that s — q > —1), and hence J1 dyF* (J - 2) ys < Cq jf' dy Q - 2^ ys < Cq jf' dy ys-« Consider the case d = 1, s > 1/2. Using F(t) > 0 for all t > 0 and F(t) = 00, Vt > 2/tt we have F*(y) = sup(ty-F(t))< sup (ty - F(t)) <-y. t>0 0 1/2 we have 1 2 /2 \ 1 4 />1 < 00. dyF* ( - - 2 ) ys" < J dy- ^- - 2J ys" < - / dyys~ < 00. □ In the special case s = 1 (sum of negative eigenvalues), the Lieb-Thirring inequality is equivalent to a kinetic inequality for orthonormal functions in L2(IRd) which will be useful to study large fermionic quantum systems. -\ Theorem (Lieb-Thirring Kinetic Inequality). Let d > 1. Let {un}n be an orthonormal family in L2(M.d) and define the density p{x) = ^2n \un(x)\2. Then ^2 J |Vii„(x)|2dx ^ Kd Jp(x)1+1ddx. 40 CHAPTER 3. SEMICLASSICAL ESTIMATES Moreover, the best constant Kd > 0 in the kinetic inequality is related to the best constant Li,d for the sum of negative eigenvalues of —A + V as = i. Remark: The Lieb-Thirring conjecture on the optimal value L\^d is equivalent to Kd = Kf = —^— ■ ^—^ when d > 3. d d + 2 \Bíň Here \BA is the volume of the unit ball in id Proof. The kinetic inequality can be proved directly using Rumin's method (see an exercise). It remains to prove the relation between the best constants Kd and L\^d. This follows a standard duality argument and Young's inequality (c.f. Legendre transform) aP ( , bq\ 11 — = sup (ab--, Va > 0, Vp,q>l, - + - P b>o V q J p q Assume that the operator — A +V has negative eigenvalues p\ < p2 < ••• with eigenfunctions ui,u2,... By the LT kinetic inequality we have V / \Vun\2>Kd í p1+~d, p(x) = V u„ (x)\2. Therefore, J2»n = yZ(un,{-A + V)un) = V f \Vun\2+ [ Vp n jRd jRd >Kd [ - / \V-\p. Using Young's inequality we find that KdP{x)1+ld - \V{x)_\p{x) > -L14\V{x)_\1+d'2, Vx G where 3.2. LIEB-THIRRING INEQUALITY 41 Thus XV > / (KdP1+^ - \V-\p) > -L14 [ \V.\1+d/2- „ jRd V ' JRd Consequently, J>„| L\4 (as Li^ is the best constant). By the choice of L\4, we get ((-N1+f(HH1+^- Reversely, consider any orthonormal family {un}n in L2(M.d) and denote p{x) = ^2n \un(x)\2. Since Young's inequality is sharp, we can choose V(x) = —CQp{x)2/d with an appropriate constant cq > 0 such that KdP1+ld-\V.\p = -L14\V.\1+d/2 where On the other hand, by the LT inequality for the sum of negative eigenvalues of —A + V and the min-max principle, V / \Vun\2+ [ Vp = J2(un,(-A + V)un)>Tr(-A + V)->-L1,d [ \V.\1+d/2. n JRd JRd n JRd Therefore, thanks to the choice of V, W |V^|2>-/ Vp-L14[ \V.\1+d/2 = Kd [ n JRd JRd JRd JRd Thus Kd < Kd since Kd is the best constant in the LT kinetic inequality. By the choice of Kd, we get 42 CHAPTER 3. SEMICLASSICAL ESTIMATES In conclusion, the best constants and L\^ satisfy = i. □ Exercise. Let d > 1. Let {wn}^Li C H1^"1) fre an orthonormal family in L2(M.d) and define pix) = ^2^=1 \un{%)\2- Use Rumin's method to prove that N f f 2 ^2 / \Vun(x)\2dx ^ Kd I p(x)1+^dx. n=1rd rd Here the constant Kd > 0 depends only on d. 3.3 Birman-Schwinger Principle In this section, we discuss an alternative approach to study the bound state problem for the Schrodinger operator —A + V(x) on L2(lRd). It is convenient to assume V < 0 and denote U = —V > 0. Our starting point is the following reformulation of the eigenvalue problem. Theorem (Birman-Schwinger principle). Let d>l. Let 0 < U G Lp(Rd) + Lg(Rd) with °o > p, q > max(l, d/2) if d ^ 2 and oo > p, q > 1 if d = 2. Recall that we can define —A — U as a self-adjoint operator on L2(M.d) with the quadratic form domain i/1(lRd) and cress(—A — U) = [0, oo). Then for all E > 0: (i) — E is an eigenvalue of —A — U if and only if 1 is an eigenvalue of KE = y/U(x)(—A + E) 1y/U(x) (with the same multiplicity). (ii) The number of eigenvalues < —E of—A — U is equal to the number of eigenvalues > 1 ofKE. Moreover, KE is a positive compact operator on L2(M.d). Proof. Step 1. We prove that if — E < 0 is an eigenvalue of —A — U(x), then 1 is an 3.3. BIRMAN-SCHWINGER PRINCIPLE 43 eigenvalue of Ke- Consider the eigenvalue equation (-A-U)f = -Ef, for some 0 ^ / G L2(rd). Then we can write (-A + E)f = Uf f = (-A + E)~1Uf Vuf = Vu(-A + E)-1Vu(Vuf). Thus = KEVUf. To conclude that Ke has eigenvalue 1, we need to show that 0 ^ VUf G L2(IRd). Since 0 ^ / G L2(Md) and -A + £ > E > 0, we have Uf = (-A + E)f^0. Consequently, y/TJf ^ 0. Moreover, since / is an eigenfunction of —A — U, it must belong to the quadratic form domain H1(Md). Hence, 'Uf\\i>= / U\f\2 g (i.e. g = VUf and / = (—A + E)~1\/Ug), we also obtain that the multiplicity of eigenvalue — E of —A — U is the same with the multiplicity of the eigenvalue 1 of KE. Step 3. We can write KE = BB* with B = ^U{x){-A + E)~xl2. Since B is a compact operator on L2(Rd) (see an exercise below), we conclude that KE is a positive compact operator on L2(Rd). Therefore, by Spectral theorem, it has eigenvalues Ai(£) > \2{E) > limA„(E) = 0. Note that for every n G N, E \—> \n(E) is decreasing and continuous. To see the monotonicity, note that E \—> KE is operator monotone: if E > E' > 0, then (-A + E)'1 < (-A + E')'1 ==» KE = VU(-A + E^VU < VU(-A + E'^VU = KE,. Hence, \n(E) < \n(E') by the min-max principle (in fact, —\n(E) is the n-th min-max value of —KE). On the other hand, if E > E' > 0, then we also have E(-A + E)-1 > E'(-A + E'y1 ==» EKE > E'KE,. Thus E\n(E) > E'\n(E') by the min-max principle again. Combining with \n(E') > \n(E), we conclude that E \—> \n(E) is also continuous. By a standard counting argument combining the one-two-one correspondence in Steps 1&2 and the monotonicity/continuity of \n(E), we find that the number of eigenvalues < E of —A — U(x) is the same with the number of eigenvalues > 1 of KE. It is easiest to see via a Figure. □ 3.3. BIRMAN-SCHWINGER PRINCIPLE 45 Exercise. Let d > 1. Let 0 < U G Lp(Md) + L9(IRd) urat/i oo > p, q > max(l, d/2) i/ d ^ 2 and oo>p,q>lifd = 2. Prove that ^JU{x){—A + E)~1/2 is a compact operator on L2(Rd). Historically, the Birman-Schwinger principle was used by Lieb and Thirring to prove their inequality Tr|(-A + V0_|s 0 when d > 2 and s > 1/2 when d = 1 (it does not work for the critical cases s = 0, d > 3 and s = 1/2, d = 1). In the following, to illustration of the usefulness of this approach, we will represent • The original proof of Lieb and Thirring in the physically interesting case s = 1, d = 3; • A proof of the existence of bound states with any negative potential in d < 2. The original proof of the Lieb-Thirring inequality for s = 1 and d = 3: Tr|(-A +V0-| < Li,3 / JR.3 It suffices to consider the case U = —V > 0. By the layer-cake representation: roc Tr|(-A-*7)_| = / A/"(—A — U + E)dE Jo where A/*(—A — U + E) is the number of negative eigenvalue of —A — U + E (which is the same to the number of eigenvalues < — E of —A + V). By the Birman-Schwinger principle, A/*(—A + V + E) is equal to the number of eigenvalues > 1 of KE = VU{x)(-A + E)-1 yfujyj. Consequently, this number is bounded by the Hilbert-Schmidt norm of Ke- Note that Ke has the kernel 1 KE(x,y) = v^G^i-j/)^, GE(k) = \27Tk\2 + E~ Thus A/"(-A - U + E) < \\KE\\ls =11 U(x)\GE(x - y)\2U(y)dxdy. 46 CHAPTER 3. SEMICLASSICAL ESTIMATES Using the Cauchy-Schwarz inequality and Plancherel's theorem we have A/"(-A -U + E)< ff \GE(x - y^^^^dxdy /jf,|2)(/j^)-(l^)(I(^FW* This bound is not good enough for inserting to the layer cake representation. But we can adjust it by shifting U h> {U — E/2)+: M(-A -U + E)= M(-A - (U - E/2) + E/2) < M{-A - (U - E/2)+ + E/2) < CE~1/2 f dx{U{x) - E/2)2+. Thus we conclude Tr|(-A-£7)_| = / d£yV(—A — U + E) Jo 0. By the Birman-Schwinger principle, — E < 0 is an eigenvalue of —A — U if and only if 1 is an eigenvalue of KE = VW){-A + E)-1 ^U{x). Since Ke is a non-negative compact operator on L2(m.d), the norm operator X1(E) = \\KE\\ is its largest eigenvalue. Let us prove that there exists E > 0 such that \i(E) = 1. 3.3. BIRMAN-SCHWINGER PRINCIPLE 47 Recall that E i—> Xi(E) is decreasing and continuous. Moreover, clearly lim Ai(E) = 0. e—J-oo (We can show that lim^^^ ||K£||Hs = 0). On the other hand, let us show that if d < 2 and 0 0+ we have, for any normalized function ip G L2(M.d), = ^\/c7, (-A + E)"V\/C7) = y |2ttA;|2 + E ' JRd |2ttA;| Here we have used Lebesgue Monotone Convergence theorem. Note that when d < 2, the function \k\~2 is not integrable at 0 G Rd. On the other hand, since 0 < U ^ 0, we have. ipyU(k) is continuous (we have ip\/U G L1(Md) when ip G L2 and [/ G L1). Thus in summary, E Xi(E) is continuous and lim Ai(E) = 0, lim A1(£') = oo Thus there exists F > 0 such that \i(E) = 1. Then —F is an eigenvalue of —A — U. □ Remark: As said, the Birman-Schwinger principle alone is not enough to derive the CLR bound. When d = 3, the Yukawa potential is given explicitly: ^ 1 e~VE\x\ GE(k) = i-T-r:-— => Ge(x) = ——:—j-- v ; \2irk\2 + E v ; 4tt|x| Hence, from the above analysis we find that JV(-A-U+E) < \\KE\&S =11 U(x)\GE(x-y)\2U(y)dxdy < [[ U(x)U(y) dxdy (47r)2\x — y\2 for all E > 0. Thus we have the Birman-Schwinger inequality for the number of all negative 48 CHAPTER 3. SEMICLASSICAL ESTIMATES eigenvalues: A/"(—A -U)< —^ [ [ (4tt)2 JJr3 \x — y\2 Recall the Hardy—Littlewood—Sobolev inequality: for p, q > 1, 0 < r < d saatisfying 1 1 r -+-+-=2 p q a we have Thus we can estimate further f{x)g{y) \x — y\r dxdy < CdyS\\f\\LP\\g\\Lq. Af(-A-U) C are uniformly bounded and vanishing at infinity, then f(x)g(—iV) is a compact operator on L2(lRd). This property can be extended to Schatten spaces. /-—-\ Definition (Schatten spaces). Let 1 < p < oo and let Jif be a Hilbert space. The Schatten space &p(Jif) contains all bounded operators A : Jif —> Jif such that \\A\\6p = (Tr(\A\p))p < oo, \A\ = ^TaFA. We denote by &oc(Jif) the space of compact operators, with the operator norm \\A\\. Thus &p{3t) C &q{3t) ifpi of Jff, we have Ti(A) = Aun) < oo. n>l The value of Ti(A) is independent of the choice of the basis {un}. Moreover, we have the cyclicality of the trace: if AB and BA are trace class, then Tr(AB) = Tt(BA). • &2(J$?) is the space of Hilbert-Schmidt operators. This is a Hilbert space with the inner product {A,B)62=Tr(A*B). When Jf? = L2(Q,,fi), any operator A G 62(^) has a kernel KA G L2(£l x £1, [i x fi) such that (Af)(x)= [ KA(x,y)f(y)dfjL(y), V/G L2(Q). Jn The mapping A \—> KA is a unitary operator from L2(£l) to L2(£l2), in particular: Il^ll62 = II-^aIU2- • In general, when A is a compact operator on J4f = L2(Q,yu), we have the spectral decomposition A = ^ K\Un)(Vn\ n>l with {un}, {vn} orthonormal families in Jrf? and A„ G R, \n —> 0 as n —> oo. The kernel of A is KA(x,y) = y^\nUn{x)vn(y). n>l With this convention, the trace of A can be computed by the diagonal part of its kernel Tr(A) = / KA(x,x)d/i(x) = y~] Xn{vn,un)L2 Jn n>l (which is well-defined if A is trace class). From the spectral decomposition, we also obtain the polar decomposition A = U\A\ with U a unitary operator on J$?. • The Schatten space &p satisfies properties similar to Lp spaces. They are Banach 50 CHAPTER 3. SEMICLASSICAL ESTIMATES spaces. Moreover, if p, q > 1 and 1/p + 1/q = 1, then we have Holder inequality \\AB\\6l < \\A\\6p\\B\\eq. More precisely, \\A\\6p= sup |Tr(AB)|= sup \{A,B)62\. \\B\\eq=l \\B\\eq=l Exercise. Prove Holder inequality: if p,q > 1 and 1/p + 1/q = 1 then \\AB\\6l < \\A\\ep\\B\\6q. Hint: You can use the spectral decomposition. Now let us come back to the operator f(x)g(—iV). A basic and very useful property is f Theorem (Kato-Seiler-Simon inequality). Let f,g£ Lp(M>d) with 2 < p < oo. Then > \\f(x)g(-zV)\\ep<\\f\\LP\\g\\LP- 4 Proof. It suffices to consider the case when 0 < /, g G L°°(lRd) with compact supports. When p = oo, it is obvious that ||/(x)£HV)||eoo < ll/IMMUoo. When p = 2, we have the exact equality ll/(^HV)||e2 = ll/IMMU- In fact, the integral kernel of f(x)g(—iV) is K(x,y) = f{x)g{x-y) where g is the inverse Fourier transform of g. Therefore ||/(*)<7(-iV)||!a = / / l/WI2l^-y)|dxdy=||/|||2|| C such that lpw ■= sup (t\{x : \f(x)\ > t}\p) < oo. T>0 V / Remarks: • Clearly the weak-Lp is smaller than the usual Lp norm: lp = [ \f\P> sup / |/|* > rp\{x : |/(x)| > r}| = ||/||^, jRd r>0 7{|/|>r} We know that Ix]"1 ^ Lp(Rd) for all p, but Ix]"1 G Lf,(lRd) since t\{x : > t}|^ = t\{x : |x| < r_1}|^ = \Bi\^. The expression ||/||lp define a quasi-norm because instead of the triangle inequality we only have + r of |+L| = yA*A. Obviously, we have \\A\\epw _ ||-*4||ep- The following deep result is interesting in its own and will imply the CLR bound. f Theorem (Cwikel's theorem). If f E Lp(lRd) and g E Lvw \\f(x)g(-zV)\\e.p,w 3 and 0 < U G Ld^2(Rd). By the Birman-Schwinger principle, for any E > 0, the number of eigenvalues < — E of — A — U(x) is equivalent to the number of eigenvalues > 1 of KE = V^R(-A + E)~1\/U{x) = ^U{x)g2E{-iV)^U{x) with gE{p) = i\p\2 + E)~xl2. Consequently, M(-A-U + E) < \\KE\\1l* = \\VU{x)9E(-iV)\\d6du. Using Cwikel's theorem and the uniform bound gs(p) _ \p\ 1 we find that \\VW)9E{-iV)\\6d,w < C\\^/U\\L4gE\\Li < CWUW1^. 3.4. KATO-SEILER-SIMON AND CWIKEL'S INEQUALITIES 53 Thus Af(-A-U + E) 0, we conclude that Af(-A - U) < C\\U\\d^/2. □ Proof of Cwikel's theorem. Assume that /, g > 0 and ||/||lp = IMIlS, = 1-Step 1. We decompose / = $>' Ux) := /(x)l(2-1 < f(x) < 2"), raGZ £ = := ^(a;)1(2""1 < 9(x) < 2"). raGZ Then where X := f{x)g{-iV) = E fn{x)gm{-iV) = Ak + Bk Ak = E fn{x)9m{-iV), Bk = E fn(x)gm(-iV), Mk G Z. m+nk By the Cauchy-Schwarz inequality, \X\2 = (A*k + S*)(Afc + Bk) < 2(A*kAk + B*kBk) = 2(\Ak\2 + |5fc|2). Step 2. We prove that Pfc||<2fe+1, VkeZ. For any normalized functions u,v G L2(Rd), by the Cauchy-Schwarz inequality we have \{u,Akv)\< E \\fnu\\\\gmv\\ =^2^2\\fnu\\\\ge_nv\\ = ^2^112-^11112-^-^11 ^~m^)2 ^ L which follow from the facts that the functions {2~nfn}n are < 1 and have disjoint supports, and the same for {2~mgm}m. Step 3. We prove that Tr(|Sfc|2) < C2(2"p)fe, VA; G Z. Indeed, it is straightforward to see that Tr(\Bk\2) = Tr(B*kBk)= ^ E Tr (9rn(-iV)fn(x)fn>(x)gm>(-iVJ) m-\-n>k m'+n'y-k = E Tr {fn(x)fn'(x)gmr(-iV)gm(-iV)^ m-\-n>k m'+n'y-k = £ Tr(|/„(x)|2|^mHV)|2) = £ \\U(x)9m(-iV)\\ m+n>k m+n>k m+n>k n£Z m>k—n Here we have used the cyclicity of the trace and the fact that {/„}„ have disjoint supports, and that {gm}m have disjoint supports. The L2-norm of gm can be controlled by the weak LP norm \\g\\Lp = 1 as follows: ll 2n~1 on its support and ||/||lp = 1 we have M\Bk\2) = '£\\fn\\l, £ ll£m|li2<£ll/Ji2 £ 2(2"p)m2P < CPT\\fn\\Ui2-p){k-n) = CP2^k [ dxV|/n(z)|22 3.4. KATO-SEILER-SIMON AND CWIKEL'S INEQUALITIES 55 0, we can choose k G Z such that £fc < £ < £fc+i = 4efc. Since e ~ 22fe, the above splitting argument tells us \X\2 0, Tr(Fe) < Cpe1_p/2. By an exercise below (with \X\2 = A) we find that \X\ is a compact operator and its eigenvalues Ai > A2 > ... satisfies K < Cpn-1/p, Vn > 1. This implies the desired inequality ||X||^ = suprp|{n G N : A„ > r}| < suprp|{n G N : CprTllv > t}\ < Cp. P'W r>0 r>0 □ Exercise. Let A > 0 be a self-adjoint operator on a Hilbert space. Let 00 > q > 1. Assume that for every e > 0, we have the operator inequality A 0, Tr(B£) < e1"9. Prove that A is a compact operator and its eigenvalues Ai > A2 > ... satisfy \n < Cn-1/q, Vn > 1. Chapter 4 Wey 1's law Weyl's law states that the semiclassical approximation (recall £_ = min(£, 0)) Ti\{-A + V{x))_\s « f f \{\2irk\2+ V{x))_\sdkdx =-Lf4 [ \V{x)_\s+d/2dx becomes correct in the strong coupling regime V i—» XV with A > 1. In principle this result holds for all d > 1 and all s > 0. Moreover, the result for one s implies the result for all others s > 0 via the layer cake representation (we will come to that). 4.1 Coherent States In this section we discuss a very general method to connect the Schrodinger operator —A V(x) and its phase-space representation. The idea goes back to Schrodinger (1926). Definition (Coherent States). Take G G Cc°°(Md), G{x) = G{-x), \\G\\L2 = 1. For every (k,y) G Rd x Rd, we defined the function Fk:V G L2(Rd) by Fk:V(x) := e2mk-xG(x -y), Vx G Rd. Note that ||FfcJ|L2(IRd) = 1 for all (k,y) G Rd x Rd. The key feature of the coherent states is that they provide a partition of the identity on L2(Rd) in terms of the phase space Rd x Rd. 56 4.1. COHERENT STATES 57 Theorem (Resolution of identity). We have I I \Fk,y){FkJdkdy = lL2 Jm.d Jm.d namely for all u G L (R ) we have \{Fhy,u)\2dkdy = M||2. Moreover, \{Fk,y,u)\2dk = (\G\2*\u\2)(y) \{Fk,y,u)\2dy = (\G\2*\u\2)(k). Proof. For any u G L2(Rd), by Plancherel theorem and G(x — y) = G(y — x) we have \{Fk,y,u)\2dk = e2mk'xG(y — x)u(x)dx dk = G(y--)u(m dk = \G(y - x)\2\\u(x)\2dx = (\G\2 * \u\2)(y). Consequently integrating over k G M.d and using ||G||L2(Rd') = 1 we obtain \{Fk:y,u)\2dkdy = I I \G(y - x)\2\\u(x)\2dxdy = \\G\\2L2\\u112 iL2 — ll"llL2- The other identity is left as an exercise. □ Exercise. Prove that for all u G L2{Rd) and k G Mr, \{Fk,y,u)\2dy=(\G\2*\u\2)(k) Now we turn to the analysis of Schrodinger operators. We have the following exact phase-space representation for the kinetic and potential operators. 58 CHAPTER 4. WEYL'SLAW Theorem. We have the quadratic form identities -ARä= I I \2Trk\2\Fk,y)(Fk,y\dkdy-\\VG\\2L2 and (V*\G\2)= J J V(y)\Fk,y){Fk,y\dkdy. Proof. Kinetic term: we prove that if u E H1^*), then [\Vu\2= [ [ ^TrfcH^^l^dy-llVGIIi.llwlli,. ■J JRd JRd Rd Using r\(Fk,y,u)\2dy=(\G\2*\u\2)(k) = / \G(k - q)\2\u(q)\2dq we have r |27rA;|2|(Ffc^,ii)|2dA;dy = f f \2irk\2\G{k - q)\2\u{q)\2dqdk JRd JRd = (2tt)2 / / \p + q\2\G(p)\2\u(q)\2dqdp = JRd JRd = (2tt)2 / / {\q\2 + \p\2+2p-q)\G(p)\2\u(q)\2dqdp The first term is (2tt)2 / / \q\2\G(p)\2\u(q)\2dqdp = ||G|||2 / |2tt9|2\u(q)\2dq = \\Vu\\2L2. JRd JRd J Rd Similarly, the second term is (2tt)2/ / b|2|G(p)|2|%)|2d^=||VG|||2||^|||2. JRd JRd 4.2. WEYL'S LAW FOR SUM OF EIGENVALUES The cross term vanishes because of the symmetry = \G(-P)\: 59 p ■ q\G(p)\2\u(q)\2dqdk = / p\G(p\2dl • / q\u(q)\2dq I =0 =0 Potential term: we prove that if u G L2(Rd), then (V * \G\2)\u\2 = [ [ V(y)\(Fk^u)\2dkdy. Recall r\{FKy,u)\2dk = (\G\2*\u\2)(y)= f \G(x-y)\2\u(x)\2dx. Here we have used G(x — y) = G(y — x). Hence, V(y)\{Fkty,u)\2dkdy= [ [ V(y)\G(x - y)\2\u(x)\2dxdy = [(V * \G\2)\u\2. □ 4.2 Weyl's law for sum of eigenvalues In this section we focus on the case s = 1, which is relevant to the ground state energy of Fermi gases. Theorem (Weyl's law). Let d > 1, V G L^oc(Rd) with p > max(l, d/2) and V- G L1+2(]Rd). Then in the limit X —> oo: Ti\(-A + \V)_\ =Lfd f |AVl|1+i+ofA1+i). ' JWLd V ' Proof. General strategy: Take a radial function 0 < G G C^°(lRd) such that ||G||£,2 = 1. Denote V = \G\2 * V. For any normalized function u G L2(Rd) we have the phase-space representation (u,(-A + \V)u)= [ [ (\2nk\2 + \V(y))\{Fkty,u)\2Mdy-\\VG\\2L2. 60 CHAPTER 4. WEYL'S LAW Therefore, for any orthonormal family {un}^=1, N ^ „ „ N VK,(-A + AyK)= / / (|2tt/c|2 + AV^(y)) V |(Ffc,„ ^) |2dMy - iV||VG|||2. n=l jRd jRd n=l The key observation is that for all (k,y) G Rd x lRd, by Bessel's inequality N 0^22\{Fk,y,un)\2 < ||FfcJ|2 = 1. n=l Therefore, by the bathtub principle N [ [ (\2irk\2 + XViy^^TliF^^^dkdy > [ [ [\2irk\2 + \V(y)]_dkdy ■JRd JRd i=1 JRd JRd = -LfJ \W-M. ■JRd Consequently, if A/* (—A + XV) < oo, then we obtain the lower bound Ti{-A + XV)_ > -Lf4 f |AV_|1+5 -M{-A + XV)\\VG\\2L2. JRd This will lead to the desired lower bound Tr(—A + xv)- > -Lf4 f |AV_ jRd |1+^+o(A1+i)A^ provided that we can • Replace V = \G\2 * V by V, namely take \G\2 —^ So. The difference is controlled by the Lieb-Thirring inequality. • Show that A/"(-A + AV0||VG|||2 < A1+l If d > 3 and VI G Ldl2, then it follows from the CLR bound A/"(-A + XV) 3 and VI G L1+^(Rd) n lJ(IRd). By the min-max principle, we can assume V = V_ < 0. Assume that —A + XV has N eigenvalues with orthonormal eigenfunctions {wj}^. By the CLR bound, N£Cd [ \XV.\l N i=l N N N We need to prove that Tr((-A + XV)-) = J>„ (-A + \V)ui) > -Lf4 J |AV1|1+^ + o(A1+i)A^co. Let us decompose N (-A + \V)ui) = £ (ui, ((1 - e)(-A) + Xv)Ul) + ^ (u^ (e(-A) + X(V - V))Ul i=l i=l i=l with a parameter e G (0,1) and V = G2 * V with a radial function 0 < G G C^°(Rd) \\G\\L2 = 1. Recall that Fk,y(x) := e2mk-xG(x - y). For the first term, we have the coherent state identity N _ f f N y2(Ui,((l-e)(-A) + v)Ui)= / / ((l-e)\2irk\2 + XV(y)) V |(Ffc,„^)|2dMy l=l l=l -N(l-e)\\VG\\2L2. Using the uniform bound N 0 < KF*.v>ui>l2 < HF^ll2 = !> V(kiV) ^RdxRd %=i we obtain N ((1 - e)|27rA:|2 + XV (y)) ^ |(Ffe,y,^)|dMy i=i 62 CHAPTER 4. WEYL'SLAW >l I [(l-e)\2irk\2 + \V(y)]_dkdy = - ^ I |AVl|1+i JWLd JWLd \l ~ £) 1 JWLd For the second term, we use the Lieb-Thirring inequality to get the lower bound N ~ r r ~ E (£(-A)+x(v - v^h) ^ JRd \x(v - v)-\1+i- i=i Combining with the above upper bound on N, we conclude that W-) > "A I |AV.r«-^ I |A(V-V).|H-Ci||VG||i, / Therefore, This holds for every e G (0,1) and V = G2 * V. Replacing G by Gn(x) =nd/2G1(nx), n>\ for a fixed function G\ we find that \Gn\2 * V —> V strongly in L1+d/2(IRd) asn-)oo (recall that we are assuming V = VI G L1+d/2). Thus hminfA-(1+d/2)Tr((-A + AV0_)^-7-^ / for all e G (0,1). Finally, sending e —> 0 we obtain the desired lower bound liminf A"(1+d/2)Tr((-A + AV0_) > -Lfd [ A^oo ' JKd Step 2: Lower bound in the general case d > 1 and VI G L1+^(IRd). Let us explain how to remove the restriction d > 3 and VI G L^(IRd). Removal of the restriction on regularity. First, the additional regularity condition on V can be removed easily using the Lieb-Thirring inequality. To be precise, let us assume that we have proved the desired lower bound liminf A-(1+d^ Tr((-A + XV)-) ^ -Lfd f A^oo ' JRa \v\1+-i 4.2. WEYL'S LAW FOR SUM OF EIGENVALUES 63 for all V- G L1^) n L°°(lRd). Now consider a general V satisfying only V- G L1+i(Rd). Then for the lower bound, we can focus on the case V = V_ < 0. Take a sequence {V^}^^ such that Vn —> V strongly in L1+%(M.d). Taking e G (0,1), we can split -A + XV = (1 - e)(-A) + XVn + e(-A) + X(V - Vn) By the min-max principle (see an exercise below), we have the lower bound Tr((-A + XV).) > Tr ([(1 - e)(-A) + Ay„] J + Tr ([e(-A) + X(V - Vn)\ J. By the assumed lower bound for functions in L Pi we have lim inf A"(1+d/2) Tr (\{1 - e) (-A) + XVn a—s-oo V L > rcl (1 - e)d/2 \V By the Lieb-Thirring inequality lim inf A"(1+d/2) Tr a—too e(-A) + X(V - Vn) > Fd/2 \V-Vn\1+*. Thus lim inf A"(1+d/2) Tr((—A + XV).) > -n /2 a=oo (1 - e)d'2 \V M Fd/2 W-Vn\1+t Sending n —> oo, and then e —> 0 we obtain the desired lower bound for Tr((—A + XV). Exercise. Let A be a self-adjoint operator on a Hilbert space such that A_ = At(A < 0) is a trace class operator. Prove that Tr(A-) = inf Tt(Aj). 0 0, we can write Tr((-A+A1/)_) = Tr((-A+A1/)1(-A+A1/ < -E))+Tr((-A+A1/)1(0 > -A+AF > -£)). Note that in low dimensions, the number of eigenvalues < — E of —A + XV is bounded (see an exercise below). Therefore, we can prove Tr((-A + XV)1(-A + XV < -E)) > -Lf4 [ \XV_\1+^ + o(A1+i) JWLd by repeating the proof in Step 1 and replacing the CLR bound by A/"(-A + XV + E) < CdE^ f \XV\2. jrd This contributes to the error o(A1+^) if we choose E such that Z^A2«A1+d/2 « E»A^. On the other hand, using the obvious operator inequality A\{Q> A>-E)>-E1-S\A_\S1 Vs G (0,1) with A = —A + XV and the Lieb-Thirring inequality we have Tr((-A + A1/)l(0 > -A + XV > -E)) > -CdE1'3 f \XV\s+%, Vl/2 < s < 1. 7Kd This contributes to the error o(A +2) if we choose E such that £i-*As+i < A1+i « E < A. 2-d Thus eventually we choose X ^> E ^> X4~d and s G (1/2,1) arbitrary. This conclude the proof of the lower bound for d < 2. Exercise. Let 3 > d > 1 and V G L2(lRd). Prove that for every E > 0 Af(-A + V + E)CG J V_M+Ld 04 [ \V-\% (V *G2 — V) dy. JWLd We can replace G by Gn(x) = nd^2Gi(nx) for a fixed function 0 < G\ G with ||Gi||L2(Rd) = 1. Since V G L1+i, we have V * G2n —> V strongly in L1+i(lRd) when n —> oo, and hence by Holder inequality / \V_\Hv*G2n-V)dy <\\\V_\i\\ ,\\V *G2n-V\\ , ■jrd = ||V_||2 d\\V*Gl - V\\ 1+d ->■ 0 asn II llLi+fll ™ ML1+2 OO. Thus we obtain the desired upper bound lim sup A"(1+d/2) Tr [-A + XV]_ < -Lf4 Í |V1 \—>CG J Step 4: Upper bound for V+ G L^oc(Rd) with some p > max(l,d/2), VI G L1+2(Rd) 4.2. WEYL'S LAW FOR SUM OF EIGENVALUES Let us explain how to remove the restrictions V+ G L1+^(M.d) and V- G L^(M.d). 67 Removal the restriction V+ G L1+2(Rd). Take V+ G L^oc(Rd) with p > max(l,d/2). For a technical reason, we assume also that V- G L°°(lRd) and V- has compact support. Then the proof in Step 3 gives us limsup A"(1+d/2) Tr [-A + \V]_ < -Lf4 f vl+^ + L$d [ \V-\$ (V * G2 - V) dy. Rd We can decompose / |Vl|i (V*G2-V) = [ \V_\1 (V_*G2-V_)dy+ [ \V_\1 (V+* G2) dy JWLd JWLd JWLd = [ \V-\$ (VI *G2 -VS)dy+ [ (|Vl|i *G2)V+dy JWLd JWLd Again, we replace G by Gn{x) = nd/2Gi{nx) for a fixed function 0 < G\ G with ||Gi||L2(Rd') = 1. Then similarly to Step 3, using V_ G L1+2 we have lim n—>oo / |Vl|i(U_*G2-Vl)dy = 0. JWLd Since V- and Gi have compact supports, the supports of V- and \V-\^ *G2l are contained in a ball Br independent of n. Moreover, since V- G L°° we have \V-\i *G2l —> \V-\d^2 strongly in any Lq(Be) with 1 < q < oo. Since V+ G Lfoc, we have V+ G Lp(Br), and hence by Holder's inequality \V-\W+ = 0. lim / (\V-\U G2n)V+ = lim / (\V.\U G2n)V+= [ Thus we get the desired upper bound lim sup A"(1+d/2) Tr [-A + \V]_ < -Lf4 f V A—>oo J Removal the restriction in VI. Now we assume only V- G L1+2(]Rd). For every R > 0, we consider VR = V+ + V-1(\x\ < R)l(\V-\ < R). 68 CHAPTER 4. WEYL'SLAW Since V < Vr, by the min-max principle and the Tr((-A + XV)-) < Tr((-A + XVR)-). Moreover, (VR)- = V-l(\x\oo A—>oo J WLd Then by Monotone Convergence Theorem, limsuPA-(1+d/2)Tr[-A + XV}_ < -Lfd lim f |(V^)_|1+^ = -Lfd [ V_\1+-2 This completes the proof of Weyl's law for the sum of negative eigenvalues. □ In the above proof of the upper bound, we have used the standard choice of the trial operator !■= / / \Fk,y){Fk,y\t(\27rk\2 + XV(y) < 0)d£xfy. JWLd JWLd We can also use a modified trial operator which simplifies the computation a bit. f Exercise. Let d > 1. Assume that VI G L1+i(Rd) and V+ G Lploc{Rd) with p > max(l,d/2) if d ^ 2 and p > 1 if d = 2. Let Fk^y(x) = e2mk'xG(x — y) with a radial function 0 < G G C%°(M.d) satisfying ||G||L2(Rd) = 1 and define the operator on L2(Rd) !■■= I I \Fk,y)(FkJt(\2irk\2 + X(G2*V)(y) + \\VG\\2L2 < 0)d£xfy. JWLd JWLd (i) Prove that Tr((-A + XV)j) = -Lf4 I (AG2 * V + ||VG|||2) 4.2. WEYL'S LAW FOR SUM OF EIGENVALUES 69 ii) Using an appropriate choice of G to deduce that 1 r V_|1+l lim sup A"(1+d/2) Tr((-A + XV)-) < -Lf4 f A-^oo ' JU.d Chapter 5 Dirichlet Laplacian -\ Definition (Dirichlet Laplacian). Let be an open set in lRd. Consider —A on L2(fl) with the core domain C£°(Cl). Since —A > 0, namely (u,-Au) = J\Vu\2 ^ 0, \/u G cC°°(Q), it defines the Dirichlet Laplacian —Ad by Friedrich's extension. The quadratic form domain of — AD is denoted by Hq(TI). Remarks: If we consider —A on L2(fl) with the core domain c°°(r2), then we also have —A > 0 and it defines the Neumann Laplacian —An by Friedrich's extension. The corresponding quadratic form is H1^) which is the Hilbert space with the natural norm IMIff1^) = ll^ll!2^) + IMIi2(si)- Here the derivatives Vu = (dXlu,dXdu) should be interpreted in the distributional sense, namely (dXdu)ipdx = - I udXdipdx, Vip G cC°°(Q). In general, Hq(Q) ^ H1^) and —Ad ^ —An- The boundary matters here! In fact, by the definition of the quadratic forms: Hliti) = cc°°(ft) , H1(Q) = c°°(Q) 70 71 In practice, any function in Hq(CI) can be approximated by a function in C£°(Tl), and the latter can be think of as a function Rd —> C (extended by 0 outside tt). This density argument makes the computations on the Dirichlet Laplacian rather similar to the usual Laplacian on L2(Rd). Obviously for any u G H1(Md), the restriction u\q always belong to H1^). The reserve direction requires the smoothness of the boundary dCl: if the boundary is C1, then H\n) = {u]n : u G H\Rd)}. In this case, we also have HHP) = {u G H\Cl) :u\m = 0}. Here the trace operator u i—y u\dQ, first defined for smooth functions u, can be extended to be a continuous linear operator H1^) —> L2(dfl). We will need only the following simple fact on Hq(CI). Lemma. For any open set C Rd, if u G H1^) and supp-u CC tt, then u G Hq(CI). Consequently, if u G i/1(lRd) and supp-u CC fi, then u\q G Hq(Q). Proof. Because supp-u CC there exists a e > 0 such that supp-u + Be{0) C £1. Choose geC?(Rd), supple BtiO), [ g = l. Define gn(x) = ndg(nx). Then gneC?(Rd), supple Bn-i(0), f gn = l. ■JM.d Denote the function u : Rd —> C by u = u(x), if x G o, if x $ n 72 CHAPTER 5. DIRICHLET LAPLACIAN Then tpn := u * gn e C™(Rd), supp<^„ c supp(-u) + supp(gn) C supp(-u) + bn-i(0) CC and ipn —>■ u strongly in H1^). Thus u e Hq(CI). □ 5.1 Berezin-Li-Yau inequality Recall the Lieb-Thirring kinetic inequality: for any orthonormal family {un}^=1 in L2(Rd), N „ N |2 W \Vun\2>Kd[ = £ n=l jRd jRd n=l un[x The Lieb-Thirring conjecture states that we actually have Kd = Kf = -1— • when d > 3. d d + 2 \bx\1 Here is the volume of the unit ball in Rd. If we assume that all functions {un} are supported on a bounded set Cl, then the LT conjecture implies that N "9 kc1 / r \ 1+^ kc1 9 n=l This weaker inequality has been proved rigorously by Berezin (1972) and Li-Yau (1983), and it holds in all dimensions d > 1. Theorem (Berezin-Li-Yau inequality). Let d > 1 and let £1 C Rd be an open bounded set. For N > 1 and any orthonormal family {un}^=1 in L2(tt) with un E H^{Q), Proof. By a density argument, we can take un E C^°(£l) for all n and think of {un} as functions Rd —> C (extended by 0 outside tt). Using Fourier transform, we can write N N N ^2 J \Vun\2 = Yl J \Vun\2 = Yl J \^k\2\un\2 = J \27Tk\2F(k)dk 5.1. BEREZIN-LI- YA U INEQ UALITY 73 where N N i=i 71=1 -2-Kik-x cxt 6Jk ~2mk-x\2dx = Here we have used the fact that {un} is an orthonormal family in L2(Q) and Bessel's inequality By the bathtub principle, it is easy to see that the minimum inf J J \27rk\2F(k)dk 0 < F < M, / F = N is attained by F0(k) = \tt\tBR(k) where ball Br = Br(0) is determined by N= J F0 = \Q\\BR\ = IQWBilR* ^ Thus Z J\^un\2^ J \27rk\2\n\dk = \n\Rd+2 J \2irk\: R = N V Ml^il dk = d 4tt2 1 77=1 Br \k\ 1 and let C M.d be an open bounded set. Let \l\ < H2 < ... be the min-max values of the Dirichlet Laplacian —Ad on L2(£l). (V) Prove that N KA 7=1 74 CHAPTER 5. DIRICHLET LAPLACIAN (ii) Deduce that —Ad has compact resolvent (hence all {/in} are eigenvalues). (iii) Prove that 0 < \i\ < The Berezin-Li-Yau inequality can be rewritten in the following dual form. Theorem (Berezin-Li-Yau inequality). Let d > 1 and let £1 C Rd be an open bounded set. Let pi < /i2 < ... be the eigenvalues of the Dirichlet Laplacian — AD on L2(£l). Then for all A > 0, we have oo 22[pn-\]->-Lf4\n\\1+l 71=1 Proof. For every N > 1, we have J>77. - A) = J> " ^VA > t-fj-.N1^ -NX> -Lf4\n\\1+-i. 71=1 71=1 Here we have used Young's inequality aP bq 11 --1--> ab, a,b>0, p,q>l, - + - = 1 p q p q and the relation ((i+§)*r((-iNf - Thus we can take N = N(\) the largest index such that < A and obtain oo N(X) J> - A]_ = 22(jm-\)> -Lf4\Q\X1+i. n=l n=l Remark: Heuristically, the above inequality justifies the Lieb-Thirring conjecture TrLHWLd)(-A + V)_>-Lf4 [ JWLd □ 5.2. SUM OF EIGENVALUES 75 -A if x G Q, -OO if x 4. £1. for the hard core potential V(x) = 5.2 Sum of eigenvalues The lower bound in the Berezin-Li-Yau inequality is sharp in the limit N —> oo, namely the semiclassical constant Kf is optimal. Theorem (Weyl's law for the sum of eigenvalues). Let d > 1 and let C Rd be an open bounded set. Then the eigenvalues \i\ < /i2 < ... of the Dirichlet Laplacian —Ad on L2(Q) satisfy - K±Kri+^„,Mi+h d (2*)* i=i IH = -^N1+d + 0(^+3)^, Kcd = m V " a d + 2 \Bl\i Proof. The Berezin-Li-Yau inequality gives the lower bound (even without error). It remains to prove the upper bound N KA Y><^v1+S+O(tf1+S)^oo. 1 = 1 I I Recall that by the min-max principle, N ( N ^2^i = inf < ^(ui, -Ad^) %=i \ %=i {ui}i=i ^ -^o(^) orthonormal family in L2(fL) if < y~]\i(ui l i>l = inf < \ Xi(ui, -AdMj) {M^i C H%(Q) an ONF in L2(Q), 0 < A, < 1, A, = N i We choose the trial operator 7 = ^i\ui){ui\ using the coherent states: 1-.= I! tBR(k)t^y)\Fk,y)(Fk,y\dkdy with a ball BR = BR{0) and a set £1 CC Cl. Recall that Fk^y(x) = e2mk'xG[x — y) with a radial function 0 < G G C£°(M.d) is a radial function. Then 7 is a trace class operator on 76 CHAPTER 5. DIRICHLET LAPLACIAN L2(IRd) satisfying 0 < 7 < 1. Moreover, we can choose R > 0 such that Tr7 = \n\\BR\ = N ^=> \tt\\Bi\R = N <^=> R = —- . \M\Bi\J In particular, we have the spectral decomposition 7 = ^^^(11,1, 0 < Aj < 1, ^Al = iV, {-Uj} ONF in L Aj = iV, \uif WIN r ill jj2^^ i>l i>l We can also require suppG C Bs(0) with 5 := ^dist(Q, Qc) > 0. Then all supp-Uj C + suppG CC because for any test function

0 supported outside Cl + supp G we have Y.X\ f ^ = Tv^21) = f f tBR(k)tn(y)\{Fk,y,ip)\2dkdy = 0 l-1 Rd Rd =o (as Fk^y = e27TkcotxG(x — y) and i d tBR(o)(k)t^(y)\\VFk:y\\2L2{Rd)dkdy ^BR(0)(k)tdy)(\^k\2 + \\VG\\2L2)dkdy = \Q\Rd+2 J \2irk\2dk+ \Vt\\BR\\\VG|||2 |fc| 1 and let £1 C Rd be an open bounded set. Prove that the eigenvalues I1! _ 1^2 _ ••• of the Dirichlet Laplacian — AD on L2(£l) satisfy oo 52\vi-\]_ = -L*MX+i + o(\1+i) 1=1 A—>oo Remark: Heuristically, the formula in the above exercise is consistent with the semiclassical formula TrL2(K3)(-A + XV). « f [ {\2irk\2 + \V{x))_dkdx = -Lf4 [ in the case V = — 1 on and > 0 (even +oo) elsewhere. \WJ1+d/2 5.3 Distribution of eigenvalues Now we come to the asymptotic behavior of a single eigenvalue, which goes back to the original result of Weyl in 1911. -\ Theorem (Weyl's law for distribution of eigenvalues). Let d > 1 and let C Rd be an open bounded set. Then the eigenvalues \i\ < /i2 < ... of the Dirichlet Laplacian —Ad on L2(Q) satisfy (27T)2 I1* = 1—12|0,2 Nd + o(Nd)N \B\ | d I d 2 , 78 CHAPTER 5. DIRICHLET LAPLACIAN Equivalently, if we denote by N{X) the number of eigenvalues < X, then Remark: The above formula of N(\) is consistent with the semiclassical formula Trl(-A + A1/ < 0) « f f t(\2irk\2 + \V{x) < 0)dkdx = Lc^d [ \\V.\d/2, L^a jBi\ (2tt)< with 1/ = -1 on Q and > 0 (even +oo) elsewhere. We will derive the above theorem using Weyl's law for the sum of eigenvalues and a simple Tauberian lemma. Lemma (Tauberian). Given any increasing sequence 0 < \i\ < \ii < stants A > 0, a > 0. Then and two con- N lim A"1"" V«n = i ^ lim N~aaN = Ail + a). 71=1 Proof. Assume that N Sn = AN±+a + o(N1+a)N^. 77=1 Then for every constant s G (0,1), with A sufficiently large and m G [eN, eN + 1) we have \SN - AN1+a\ < e2N1+a, \SN+m - A(N + m)1+a\ +-- + 2e(l + s)1+a. iV-^oo £ 5.3. DISTRIBUTION OF EIGENVALUES 79 Sending e —> 0+ we obtain limsupi\r>jv < A(l + a). AT->oo Similarly, using /ijV + vn-l + ••• + pN-m+l Sn — Sn-tti un >-- m m we find that liminfiV->jv > A(l + a). Thus lim N-a(iN = A(l + a). N—too The reverse direction is left as an exercise. □ Exercise. Given an increasing sequence 0 < [i\ < /i2 _ ••• satisfying lim N~afiN = A(l + a) TV—>oc for two constants A > 0, a > 0. Prove that N lim N-x-a = A. n=l Proof of Weyl's law for the distribution of eigenvalues. In the previous section we have proved that , 2 A Kf , d (2tt)2 lim iV"1— V //i = —=3-, Kf = -—- • ^3-. Therefore, the Tauberian lemma implies that 2 Kc} ( 2\ (2tt)2 lim N-*(iN = —T-fl + 3) = 2 2- Now consider iV(A) the number of eigenvalues < A. By definition of N(\), we have fJ'N(X) < A < [lN(\)+i. Of course, when A —> oo then N(X) —> oo. Hence by the asymptotic formula of jiN for N 80 CHAPTER 5. DIRICHLET LAPLACIAN large, | -D11 d |l l \ d which is equivalent to □ 5.4 Polya conjecture Let d > 1 and let ft C Rd be an open bounded set. Recall Weyl's law for the eigenvalues Pi < p2 < ••• of the Dirichlet Laplacian —Ad on L2(ft): = , |2| [2Nd + 0(^3)^00. |-E>l| d |Q| d An important open problem is Polya's conjecture: »n >—^-N$, \/N>l ~ \B1\^\n\2/d which is equivalent to (why?) "W<^'. VA>0. The Berezin-Li-Yau inequality follows from Polya's conjecture. Clearly A (2tt)2 A 2 (2tt)2 [N 2 , (2tt)2 / 2\ 1 + 2 Kf , , 2 ^ ~ \B1\l\n\2id^ ~ \B1\2\n\2id Jo |o|2/rf v dJ However, obtaining the sharp lower bound for every eigenvalue is much more difficult. Nevertheless, using the Berezin-Li-Yau inequality we get the non-optimal bound 1 A Kf 2 (2tt)2 d 2 UAr m-N^/n-Mi \B1\iMVdd + 2 ' n=l II I -1-1 I I □ Proof of Polya's conjecture for cubes. Polya's conjecture can be verified easily for cubes. 5.4. POLY A CONJECTURE 81 By a simple scaling argument, it suffices to consider the case = [0, ir]d where the eigenvalues are given explicitly by {|x|2 = (x2 + ... +x2d)\x= (xi,xd) G Nd}. A key observation is that the number of integer points inside a ball can be controlled by the volume of the ball. More precisely, any point x = (xl5 ...,xd) G Nd} can be associated with the unit cube Qx = (xi, xi - 1) x (x2, x2 - 1) x ...(xd, xd-l). Since Qx n Qy = 0 if x ^ y and Qx C BR(0) n R^ if x G Nd n BR(0), we find that |Ndn BR(0)\ U Q* xGNdnBfl(0) <2~a\BR\ = 2~dRd\B1 Figure: Positive integer points inside a circle On the other hand, since {/in} is an increasing sequence, there must be at least N points inside \Nd n BR(0)\ with R = ^fp~^. Thus N < 2~dpl\B1 Pn > 22 „2 (2tt) I Si -Ni, VN>1. □ 82 CHAPTER 5. DIRICHLET LAPLACIAN In 1961, Polya proved Theorem (Polya). Let d > 1 and let £1 C Rd be an open bounded set. Assume that £1 is a tiling domain, namely we can cover Rd (up to a set of 0 measure) by a union of disjoint copies of £1 (each copy is obtained from by £1 up to translation, rotation and reflection). Then the eigenvalues \i\ < /i2 < ... of the Dirichlet Laplacian —Ad on L2{fl) satisfy (27T)2 2 Vn>—-r^-N*. VJV>1. Remark: A cube is a tiling domain, but a ball is not (this case remains open). Figure: Tiling by hexagonal Proof. Let us denote by ^(Q) the £>th eigenvalue of the Dirichlet Laplacian — Ad on L2(fl). Step 1. Assuming that we can put N disjoint copies {^tn}n=i °f ^ inside a large cube Q C Rd. We will prove yUfc(Q) > iikn(Q), VA; = 1,2,... Take k > 1. By the min-max principle, for every e > 0 we can find a subspace Mk(£l) C CC°°(Q) such that dimMfc(Q) = k and yUfc(Q) > sup ||V-u|||2 — e. 5.4. POLY A CONJECTURE 83 Since each £ln is a copy of £1, we find that /ik(^ln) = ^fc(^) and we can also find a subspace Mk(ttn) C CC°°(Q) such that dimMfc(Q„) = k and yUfc(Q) = pk{yin) > sup ||Vii|||2 - e. ueMfe(Qn) II"IIl2=i Note that the functions in Mk(Q,n) have disjoint supports to the functions in Mfc(Qm) if n 7^ m. Therefore, the space M = 0 Mfc(Q„) = Spanjn G |J Mfc(Q„)} C CC°°(Q) 77=1 77=1 has dim M = kN. Then by the min-max principle, PkN^) 5: SUP ll^^lllz 5: SUP SUP ll^^lll2 -~^fc(^)+£- mGM l fjLkN(Q) > ln]2'(kN)*, Vfc>l. 2 / |,„|2 84 CHAPTER 5. DIRICHLET LAPLACIAN This inequality holds for all N > 1 such that we can put at least N disjoint copies of £1 inside the cube Q C Rd. Since Q is a tilling domain, we can choose a very big cube QN C Rd such that we can put N disjoint copies of £1 inside QN and at the same time r \Qn\ 1 um "T7T7TT = 1- JV->oo N\tt\ Thus for every k > 1, we have (2tt)2 2 (2^2 7T , 2 This completes the proof. □ Remark: For a general domain Cl, by using the above proof, we find that /ife(fi) > i?(Q)2/d—^—0, Vk > 1. where f?(Q) G (0,1] is the packing density of tt, namely the largest fraction of the space ]Rd that we can cover by disjoint copies of fl. Determination the packing density of a ball is the standard packing problem. 2D Packing problem. For a disc = {x G R2 : |x| < 1}, the packing density is R(Q) = -£= ~ 0.9069... Vl2 and it is achieved by the "hexagonal packing arrangement". This optimality was proved by Lagrange in 1773 for "lattice packings", by Gauss in 1831 for "periodic packings", and finally by Toth in 1940 for the general case. 3D Packing problem. For a ball = {x G R3 : |x| < 1}, the "sphere packing problem" is more difficult. In 1611, Kepler conjectured that the optimal parking arrangement is obtained by a family of "close-packed structures", leading to R(Q) = -^-= w 0.74048... 3v 2 This optimality was proved by Gauss in 1831 for "lattice packings" and by Hales in 1998 for the general case but his proof is involving a heavy computer checking of many individual 5.5. WEYL'S CONJECTURE 85 cases. Finally, a formal proof was published in 2017 by Hales and collaborators (Forum of Mathematics, Pi). 8D & 24D Packing problems. In 2016, Viazovska published a surprisingly short solution for the parking problem in 8 dimensions. Shortly later, she and collaborators solved the problem in 24 dimensions. Figure: Circle packing and "Sphere packing" 5.5 Weyl's conjecture Recall that Weyl's law states that the number of eigenvalues < A of the Dirichlet Laplacian —Ad on L2(fl) satifies n{\) = l$d\q\\* +o(A2) A—>ooi ! _ |^(0, 1)| Weyl's conjecture (1911) states that the second order term is involving \dtt\ 1 n(\) = L«d\n\\-2 - ^L^imix— + o(A—) A—>oo* (The second order term is negative, so it is consistent with Polya's conjecture). This is a hard problem. A proof of Weyl's conjecture for a class of smooth domains was given by Ivrii in 1980 (if you really love semiclassical approximation, check Ivrii's Monsterbook (2007-2019)). Proof of Weyl's conjecture for squares. 86 CHAPTERS. DIRICHLET LAPLACIAN Recall that for Q = [0,7r]2, the eigenvalues of the Dirichlet Laplacian —Ad on L2{p.) are given explicitly by {|x|2 = (x\ + x\) I x = (xi, x2) G N2}. Thus N(X) = ^(S(y/X)-4,[V\\ - 1) where the floor function |_^J is the integer part of t (i.e. [t\ < t < [t\ + 1 £ N) and S(R) := number of integer points inside -8(0, R). The Weyl's conjecture tells us that for q = [0,7r 2 N(\) = L*d\n\xi - ^L^ionix^ + o^)^ = ^X-X1/2 + o(X^)x^00 which is equivalent to S(R) = ttR2 + oiR)^. The asymptotic estimate for \S(R)—irR2\ when R —> oo is called the Gauss circle problem (1801). Hardy conjectured that \S{R) - ttR2\ < 0{R1/2+£)R^, Ve > 0. The lower bound is sharp since Hardy and Landau independently showed that the error cannot be better than 0{R}^2 ln(f?)). On the other hand, the upper bound remains open. Gauss himself managed to prove that \S(R) -ttR2\ < 2V2ttR. Sierpinski (1905) proved that \S{R) - ttR2\ < 0(^)^00, V6> > \ « 0.66666.... Huxley (2003) proved the currently best bound 131 \S(R) - ttR2\ < 0{Re)R^ 0 = — ^ 0.62981... zUo 5.5. WEYL'S CONJECTURE 87 Now let us prove \S(R) — ttR2\ < o(R) which is necessary to justify Weyl's conjecture for squares. This means we need to improve Gauss' bound, which is a nontrivial task. We start with Theorem (Poisson summation formula). If f £ C£°(M. ), then E /(*) = E /(*)• xGZd fcGZd In general, this formula holds if / is sufficiently smooth and decays sufficiently fast (e.g. / G C°° and it decays faster than any polynomial). Proof. Consider the function F(x)=^2f(x + y). yGZd Then F : Rd —> C is a periodic function (of period 1) and it can be written by the Fourier series where F(x) = E ake2mk-x fcezd Thus ak= [ F{x)e-2mk-xdx = E [ f(x + y)e-2mk'xdx JV>W ,GZd JVW = E / f{x)e-2mk-xdx = [ f(x)e-2™k-xdx = f(k). YJf{x + y) = F{x)=YJKk)e2mk-x. In particular, taking x = 0 we obtain E Kv) = E /(*)• yezd fcezd □ 88 CHAPTER 5. DIRICHLET LAPLACIAN Remark: Strictly speaking, the formula lm oo (which can be seen by integration by part). Now we prove Theorem. Let S(R) be the number of integer points inside B(0,R) C R2. Then \S(R)-ttR2\<0(R2/3)r^qo. Proof. The proof is due to Hugh Montgomery. Step 1. Note that S(R)=^1Bb(0)(x). xGZ2 We want to use the Poisson summation formula, and hence we will replace 1^(0) by smooth functions. Fix a radial function 0 < h > 0 0 we denote fT(x) = tph* lBr(0)(x) = / iph(x - y)dy. 'Br(O) 5.5. WEYL'S CONJECTURE 89 Since fr 6 C^°, we have Poisson summation formula xGZ2 fcGZ2 Moreover, clearly ^Vh-2Vl(x/h)dx Jr2 -2™(kh>y0. j Moreover, since (pi G C%°, ipl G L°° and it decays faster than any polynomial. Thus •~ r Cor1/2 \fr{k)\ = mhk)\-jMk\)< (/#[m|3/2> w-°- Take a cut-off K > 0. We have E \ln< E S^'"^1" o<|fc|i ^ E £S ^ c/1'2 Thus /,.|Af/2 - ~ hK1/2' \k\>K \k\>K 1 1 ^ „1/2 £ 1/^)1 0 we find that \S(r)-nr2\ = ^\fr(k)\-i z i \ -2 3 -1 > r \ 3+5 H \ 1 5 -1 ■2' / \ \ 6 / — At _ Figure: Two isospectral domains with different shapes 5.6. CAN ONE HEAR THE SHAPE OF A DRUM? 93 Why v constructed this way is an eigenfunction on with eigenvalue \il • First, we have Au = \m in the interior part of each triangle in fli, so obviously Av = /iv on the interior part of each triangle of r^. • Second, we have to check that v vanishes on the boundary of r^. • Third, we have to check that v is smooth inside ^2, namely it is smoothly connected on the joint boundary of the triangles of ^2- Finally, we have to check that the multiplicity of eigenvalues are the same in fli and Q2. This means that the transplantation is invertible. Fortunately, this is true for this example. On the other direction, Zelditch proved a positive answer for 2D sets which are convex with analytic boundary. An open question is that whether the convexity can be relaxed? Chapter 6 Neumann Laplacian t Definition (Neumann Laplacian). Let be an open set in lRd. The Neumann Laplacian —AN on L2(£l) is defined by Friedrich's extension via the quadratic form formula (u,-ANu) = J\Vu\2 ^ 0, Vu G C The quadratic form domain of — AN is H1^). fTO>d\ Recall that H1^) is a Hilbert space with the norm IMIff^si) = ll^ll!2^) + ll'ulll2(si)- Here the derivatives are interpreted in the distributional sense, namely (dXdu)ipdx = - udXdipdx, V

1 we have yU„(-AD) If —An has compact resolvent, then all /in(—An) are eigenvalues, and the stronger inequality /J.n(—Ad) > fj,n+i{—An) holds (called Friedlander's inequality). inf max / V-u > inf max / V-u = yu„(—Aj dimM=n IMIr2=lsi dimM=n IMIr2 = l i 96 CHAPTER 6. NEUMANN LAPLACIAN 6.1 Essential spectrum vs. Compact resolvent Unlike the Dirichlet Laplacian on any bounded set always has compact resolvent, the bound-edness of the domain is not enough to ensure that the Neumann Laplacian has compact resolvent. Putting differently, when is bounded, the embedding Hq(CI) C L2(fl) is always compact, but the embedding H1^) C L2(fl) is not necessarily compact. Trivial example: We can take = U™=1-E>„ be a union of disjoint balls. The functions are orthonormal in L2(fl) and all have JQ |V-u„|2 = 0. Thus /J.n(—An) = 0 for all n > 1. In this example, however, £1 is not connected. Example "Rooms and Passages" (Courant and Hilbert). Let C R2 be a union of a sequence of "rooms" and "passages" where • the n-th room is a rectangle of size an x bn. • the n-th passage is a rectangle of size en x bn. J 1 \ ft Hi ) 1 U iSjopc. -« —:--- fir Ln Ln c 1>1 < y %- -1 u \ r-i LI 1 K ti i Ar ■ y < bL Figure: Rooms and Passages Now the set Q is simply connected and it is bounded if ^2nbn < oo. On the n-th room, we can take the function un ~ (an&n)-1^2 and interpolate it to 0 linearly up to the middle points 6.2. EXTENSION DOMAINS 97 of the connected passages. Thus {un}n>i are orthonormal in L2(£l) and / Wun\2 < ( bn J fa v , _ e. This can be made small uniformly in n by taking en 1 and any close set S C [0, oo), there exists an open, connected, bounded subset £1 C Rd such that the spectrum of the Neumann Laplacian on L2(£l) is equal to S. Thus to ensure that —An has compact resolvent, i.e. the embedding H1^) C L2(fl) is compact, we need something more than the usual requirement of the boundedness of £1. A sufficient condition is that dtt is sufficiently smooth (e.g. C1), or more generally that is an extension domain. 6.2 Extension domains Clearly, if u G H1(Md), then u\q G H1^). More generally, if C fi, then we have the obvious restriction H1^) C H1^). Reversely, the extension is less trivial and requires some conditions on the boundary dtt. d Definition (Extension domains). Let O C R fte an open set. We call £1 an extension domain if there exists an extension operator E : H1^) —> i/1(lRd) such that (Eu)\n = u, 11 Eu 11 i2(Rd) < C|H|L2(n), WEuWhi^ < C\\u\\Hi{n). To illustrate the idea, we have f > Lemma (Extension by reflection). Consider the half-space = IRd_1 x R+ = {x = (xi,xd) : xd > 0}. For any function u : £1 —> C we define the extension Eu : Rd —> C 98 CHAPTER 6. NEUMANN LAPLACIAN by J u(x), Eu{x) = < 1 u(xi,. ifxd > 0, ifxd<0. Ifu G H1^), then Eu G H^(Rd). <* Proof. Clearly Eu G L2(Rd) and ||£n|||2(R2) = 2\\u\\L2{Q). For the derivatives, let us denote another extension E by Euix) = uix), if xd > 0, -it(xi,xd_i, — xd), il xd < 0. Then we have ' Edx.u, if j = 1,2,...,d- 1, EdXju,, if j: = d. In fact, for every test function <£> G C^°(lRd) we can write: for any j = 1, 2, d — 1 (Eu)(x)dXjip(x)dx = / u(x)dXj(p(x)dx + / uixi, -xd)dXjip(x)dx (dXju)(x)(p(x)dx - / {dX]u){x1, ...xd_1,-xd)ip{x)dx (Edx,u)(x)ip(x)dx - / (EdXju)(xi, ...xd-i,xd)tp(x)Ax (Edx,u)(x)ip(x)dx and for j = d, {Eu){x)dXd(p{x)dx = / u{x)dXd(p{x)(\x + / u{xx,xd, -xd)dXd(p(x)dx Jn Jnc (dXdu)(x)ip(x)dx - / (-dXdu)(y,-xd)ip(x)dx ft Jfic (EdXju)(x)ip(x)dx - / (EdXju)(y,xd)(p(x)dx (EdXju)(x)ip(x)dx. 6.2. EXTENSION DOMAINS 99 Hence / |V(Z^)|2 = W \EdX]u\2 + [ \EdXd JRd -=1 JRd JRd d—1 „ „ = 2^ / \dX]u\2 + 2 / \dXduf 2 _ OIL, 112 = 2 / |Vd2. Thus Eu G i/1(]R C the extension -u : Q —>■ C by uix) u(x), if x G ÍŽ+, -u(xl5 Xd_!, — Xd), if x G íž_ < 0. By repeating this procedure a few times. Moreover, note that ipu G H1(Md) if ip G C^°(lRd) and -u G H1(supp A 1 VT Figure: Extension by reflection This technique can be made general by Definition. Let Q = BRd-i(0,1) x (-1,1) = Q+UQ_U Q0 with Q+ = 5^-1(0,1) x (0,1), Q_ =5^-! (0,1) x (-1,0)^0 = 5^-1(0,1) x {0}. 100 CHAPTER 6. NEUMANN LAPLACIAN Let £1 C Rd be an open set. We say that dtt £ C1 if for every x £ dtt, there exists an open set x £ U C Rd and a bijective map 0 : Rd —> Rd such that e.e^ec1, e(u) = Q, e([/nn) = Q_, e(undn) = Q0. if ///// m w Y//A a y Figure: C1 boundary Theorem. Let Cl C Rd be an open bounded set with dtt £ C1. Then Cl is an extension domain. Proof. Since dtt is compact and C1 smooth, it can be covered by finitely many open sets Ui,Uk C Rd such that in each set U% we can find a bijective map Oj : lRd —> lRd such that Quebec1, Qi(ui) = Q, el(ulnn) = Q_ Moreover, we can find smooths function {(Pi}^=0 C C°°(lRd) • 0 < C E((piU) = (Pi(x)vi(Oi(x)), if x G Ui, 0, if x G Rd\Ui. Clearly £(tftii) G i/1^) since v^B^x)) G tf1^) and Vi ^ C™(Ui). Extension of <^ow. We simply define the extension E(■ C by E{ 0 strongly in L2(fl). The condition un —^ 0 weakly in H1^) implies that • un —^ 0 weakly in L2(£l), since i/1-norm is stronger than L2-norm; • un is bounded in i/1(Q), by Banach-Steinhaus theorem. 102 CHAPTER 6. NEUMANN LAPLACIAN Let E : H1^) —> i/1(IRd) be an extension operator. Then Eun is bounded in i/1(IRd). Thus up to a subsequence, we can assume that Eun -± g weakly in H1^) by Banach-Alaoglu theorem. By Sobolev embedding theorem, we obtain \^Eun —> 1q<7 strongly in L2(IRd), namely un -)■ g\n strongly in L2(Q). Since un —^ 0 weakly in L2(£l), we conclude that g^ = 0, and hence un —> 0 strongly in L2(tt). Since the limit is unique, this convergence holds for the whole sequence. Thus the embedding H1^) C L2(fl) is compact. This implies that (—An + 1)_1 is a compact operator. Indeed, if un —^ 0 weakly in L2(fL), then vn := (-AN + 1)"V ->■ 0 strongly in L2(fl) as follows. Since -u„ —^ 0 weakly in L2(fl) and (—AN + 1)_1 is bounded, vn —^ 0 weakly in L2(Q). Moreover, IWIfl+Sl) = (Vni(-^ + ^)vn)L2(Q) = {vn, Un)L2(Q) < \\Un\\L2(Q) \\Vn\\L2(Q) is bounded. Since H1^) C L2(fL) is compact, up to a subsequence, vn —> v strongly in L2(fl). Since vn —^ 0 weakly in L2(fL), we must have v = 0. This completes the proof. □ Remark: For any open bounded set C Rd, the embedding Hr)(tt) C L2(fl) is always compact since Hq(CI) can be always extended to i/1(lRd) "by 0 from outside". Exercise. Let d > 1 and let 1 and let C M.d be an open bounded set such that the embedding H1^) C L2(tt) is compact. Then the eigenvalues \l\ < ^2 _ ••• of the Neumann Laplacian —An on L2{Q) satisfy N R-cl V a < d N1+1d Kf = n=l ft|2/d ' d d + 2 \B, i 2 Proof. Let {"ura}ra>i be an orthonormal basis of eigenfunctions of —An- Then -AN-Un = [inUn and hence I l2(Q) \ I l2(Q) l2(q) \ / l2(q) By the min-max principle, we know that Pn = inf / |V-u|2. U±U1,...,UN-1 Jq Wuh2(n)=1 For any function / G H1^), clearly N-l n=l is orthogonal to all ui, ...,uN_1 in L2(Q). Hence a Jn 104 CHAPTER 6. NEUMANN LAPLACIAN By the definition of v, we can compute AT-1 m2-E (WW) n=l and / |V?f = [ V.f-Y{un,f)LHü)Vv r N-l N-l . = / \Vf\2-2^(Y{unJ)LHa)(vf,Vun) ) + £ K«n, />l»(n)|2 / |Vt „ AT-1 AT-1 = / \vf\2-2^[YJ(^f)m^n(f,un) ) +ElK./)wlV n=l ^ n=l Here we have used the fact that un are eigenfunctions of — AN. Thus in summary, AT-1 AT-1 for k G Md and obtain / \Vf\2-f^Pn\(UnJ)»M\2>»n( f l/r-El^'^Mn)2)' V/G^^)- ™=i V ™=i ' In particular, we can choose f(x) = e N-l n=l Integrating over /c G BR(0) C Rd we get AT-l dfc|2^fc|2|ft| - AT-l 71=1 n=l JbR JbR n=l N-l Now we choose R such that N=\Ü\\BR\ = |^!||Si|i?d « i? 6.4. LIEB-THIRRING INEQUALITY FOR NEUMANN LAPLACIAN 105 This gives J \2irk\2\tt\dk = \tt\Rd+2 J \2irk\2dk = d 4tt2 1 . , 2 Kf , , 2 BÄ |fc|fiN(N-^2 Í dk\í^rn{k)\2^j n=l JER n=l R which is equivalent to _cl N N-l „ N-l .1 , ,1 i 2 I^r n=l n=l ^BR n=l ^ The right side is > 0 because for every n = 1,2,N — lwe have hn > and Thus d/c|lslii„(A;)|2 < / d/c|lslii„(A;)|2 = / dx|(l^n)(x)|2 = / dx|ii„(x)|2 = 1. br JwLd JwLd Jn Kc\ N li^l2 n=l □ 6.4 Lieb—Thirring inequality for Neumann Laplacian Recall the standard Lieb-Thirring kinetic inequality: For every d > 1, there exists a constant Kd > 0 such that for all N > 1 and for all orthonormal functions {un}^=1 in L2(Rd), then N / |Vü„|2 ^ Kd / = Y \Mx)\2- n=1WLd WLd The following generalization will be useful f-\ 106 CHAPTER 6. NEUMANN LAPLACIAN Exercise. Letd>\. Let N > 1 and let C H1^) satisfy n ^2\un){un\ 0 is the same as in the case of orthonormal functions. This inequality can be also written in the compact form: for any trace class operator 0 < 7 < 1 on L2(Rd), then Tr(-A7) >Kd f p ■Jm.d 1+2/d 7 Here Tr(-A7) := Tr(^/-A7^/-A) and Pl[x) = \un{x)\2 if 7 = E„ K)(un\. Now we want to extend the above inequality for functions in L2(tt) with an open bounded set Q C Rd. • This inequality extended immediately to the Dirichlet Laplacian, namely if {un}n C Hq(CI) and EtT=i \un){un\ — 1 on L2(Q), then n r. „ n |2 £ J \Vun\2 > Kd f p(x) = f^ unlx The reason is that the extension Hq(CI) —> H1(Md) is trivial (we simply set u\qc = 0). The analogue for Neumann Laplacian is less obvious since the extension H1^) —) i/1(]Rd) is more complicated. Theorem (Lieb-Thirring kinetic inequality for Neumann Laplacian). Let d > 1 and let C Rd be an open bounded set with dtt G C1. Let N > 1 and let {un}^=1 C H1^) such that ^2n=1 \un)(un\ < 1 on L2{fl). Then n r r r \2 £ /\Vun\2 >Kn f p1+i - f p, p(x) = £ un(x 6.4. LIEB-THIRRING INEQUALITY FOR NEUMANN LAPLACIAN 107 Our idea is to use the extension operator E : H1^) —> i/1(IRd) (which requires dtt G C1) and then apply the standard Lieb-Thirring inequality for L2(Rd). The key observation is that the extension operator does not destroy the orthogonality too much. Lemma. Let d > 1 and let £1 C Rd be an open bounded set with dtt G C1. Let E : H1^) —> H1(Rd) be the extension operator constructed in a previous section. Then for every N > 1, if N \un)(un\ < 1 on L2(fl), 71=1 then N ^2\Eun){Eun\ Rd such that QuQ^eC1, Qi(Ui) = Q, Ql(Ul nO) = Q_, Q%{U% n dtl) = Q0. Then we use a partition of unity 1 = X]/=o fj on ^d with {Vj}/=o ^ C°°(lRd) such that 0 < (fj < 1, suppt^ CC f/j Vj 0, supp^CCRV^- We decompose j u = E ^ J=0 and extend t^-ii as follows: 108 CHAPTER 6. NEUMANN LAPLACIAN 7/ \ W Q + u y//A 1 / a y Figure: The mapping /ij = Ql 1RQl : £/i n ft Ut\Q • For j = 0, v?0u E H%(£1) C i^QR^) (by setting ( C by T^-u = -u on n ft and Tju(y) = it(/i~1(y)) with y G £/j\ft where ^ = e-^e^: ^ n ft ^ L^\ft (here R : ■ d\ n=l namely < C \g\\ Vg G L 2 /n>d \ V] / (Eun)(x)g(x)d n=l ' By the Cauchy-Schwarz inequality for complex numbers N r 2 N r J V / (Eun)g = Yj / ((pou + ^^jTjUnjg / (W^)# +Z / (lP3T3U n)g n=l n=l TV <2 [ |^(x)|2|(/(x)|2 + 2||DetJ^||Loo f (^(^(xjjH^^^x^plDetJ/i^xJldx =<2 f |^(x)|2Kx)|2 + 2||DetJ^||Loo f \^(y)\2\g(y)\2dy H1(Md) be the extension operator we discussed above. Then llE^H^^d) < C||-u„||l2(si), llE^ll^i^d) < (711^11^1(0), Vn = 1, 2,N and AT 22\Eun){Eun\ Kn /p1+2/d- □ 6.5 Weyl's law Theorem (Weyl's law for distribution of Neumann eigenvalues). Let d > 1 and let C M.d be an open bounded set with dfl G C1. Then the eigenvalues \l\ < ^2 _ ••• of the Neumann Laplacian —An on L2{0) satisfy I1* = ^l]^iN~d + o(N*)N. \B\ \ 0 small, by the inner regularity of Lebesgue measure, we can find an open set fl£ CC such that \n\n£\ < e. Since dist(r2e, flc) > 0, we can find a smooth function ip£ G such that 0 < ty?e < 1, tpe = 1 on ^e) suPP(i be an orthonormal basis of eigenfunctions of — An on L2(fl). By the Cauchy-Schwarz inequality we can bound v(Veun)\2= / \(yVe)un + Ve(yUn))\2 n Jn = I |V^£|>„|2 + / \ip£\2\Vun\2 + / (Vip£)unip£(Vi Jn Jn Jn <(l + d-r) f |V^|2K|2 + (l + <5) f |^|2|V^|2 Jn Jn < C£:S + (1 + S)/in. for all n > 1 and S > 0. Thus N N N 0£Un)\- - Ls£x u„)\2-C£xN. (1 + 5)^2^ = (1 + 5)^2 [ \Vun\2>Y [ Mfe n=l n=l JV n=l J" Step 2. Now we estimate Yln=i JQ |V(i^(E/owa) n=l JU 11 n=l JU Thus in summary, we obtain Combining with the bound from Step 1, we find that (! +s) E ^ pfe - - 1+" - n=l Step 3. Now we bound (l-^)p< / P- Using Kröger's inequality and the Lieb-Thirring inequality for orthonormal functions {un} C 6.6. POLY A CONJECTURE 113 H1^), we have K. •cl N N 77=1 n=lJV Jn 1+21 d _ N s'l ' - - ./o ./q Hence, By Holder's inequality and the choice |ri\rie| < e, P< ||l||L1+d/2(si\a)l|p|lL1+2/d(si\sis) < Cne2/(d+2)N. q\qe Step 4. In conclusion, for every e > 0 and 5 > 0 we have A N „d (X + 5) E ^ ^ - ^2/(d+2) a) 3 - CEtsN. Consequently, 71=1 (1 + 4) lim i„f iV-V- 53 ^ > *L (i - Ch^'«>)'' '. 77=1 1 1 Taking e —> 0, and then 5 —> 0, we conclude that liminfA-1-2/^^^-^-. 77=1 1 1 This completes the desired lower bound for the sum of Neumann eigenvalues, and completes the proof of Weyl's law for the distribution of eigenvalues. □ 6.6 Polya conjecture As we have seen, many inequalities change their directions when we turn Dirichlet to Neumann eigenvalues. Polya's conjecture states that (27r)2 2 yU„(-AD) >--- nd > fin+1(-AN), Vn > 1. \Bl\d \il\2/d 114 CHAPTER 6. NEUMANN LAPLACIAN Proof of Polya's conjecture for Neumann eigenvalues of cubes. Let 0 C ld be a cube. By scaling, it suffices to consider the case = [0, ir]d where the Neumann eigenvalues are given explicitly by {|x|2 = {x\ + ... + x2d)\x = (Xl,xd) G Ng}, N0 = {0,1, 2,...}. If we denote for any point x = xd) the cube Qx = (xi, xi + 1) x (x2, x2 + 1) x ...(xd, xd + 1), then \NdnBR(0)\ = |J Q xGNgnBfl(0) > 2-d\BR\ = 2-d\B1\R1. ■ X '/A Figure: Nonnegative integer points inside a circle Thus for every A > 0, the number of Neumann eigenvalues < A, which is equal to the number of non-negative integer points inside -8(0, \/A), satisfies N(X) = \Nd n 5^(0)1 > 2-d|51|A"/2 = ^^i, VA > 0. 6.6. POLY A CONJECTURE 115 Since there are at most n eigenvalues < [in+i, we have which is equivalent to u„ ii < -=-n 1. + |Si|i|fi|2/d Polya (1961) extended this result for a sub-class of tiling domains. A proof for all tilling domains was just found recently by Filonov (June 2020). f Theorem. Let d > 1 and let C M.d be an open bounded set such that the embedding H1^) C L2(tt) is compact. Assume that is a tiling domain, namely we can cover M.d (up to a set of 0 measure) by a union of disjoint copies of Q, (each copy is obtained from by £1 up to translation, rotation and reflection). Then the eigenvalues \i\ < /i2 < ... of the Neumann Laplacian —An on L2(fA) satisfy (2tt)2 2 w < -«-n 1. ^+ iSxlilfip/d Equivalently, the number of Neumann eigenvalues < X satisfies \-Bi_ (2tt) We will need the sub-additivity of \—> N(\, tt). Lemma. Let {£lj}j=i be disjoint open sets in R . Then N(X, n) < E N(X, nA, n = interior of (|J Cl^. Proof. First, we assume N(\, tt) = k, namely l^k < X < i^k+i- By the min-max principle, /ik = inf sup dim Mk=k In Vu 2 In \u\ 2 116 CHAPTER 6. NEUMANN LAPLACIAN Note that j j H\n) C X = 0 tf1^-) = {u G L2(tt) I u = £ lo,^ such that ^ G H1^) for all j}. Hence, _ . f E^LJv.i2 Pfc > Pk = mi sup dimMfe=fc k ^J=l j^j 1 1 On the other hand, for every subspace Mk C X with dimMfc = k, we can decompose j j Mfc = 0Mfcj, Mktj cH\n3), dimMk,3=£(k,j), ^£(k,j) = k. Hence, by the min-max principle for the Neumann Laplacian on each r^-, E/=iL.lv^l2 / Jn.|v«r\ sup ——j—3 > sup sup 3 > sup fin^fy). 2^=iJnM i iik> iik> inf sup /^(fcj-) (£!,•). The infimum is taken over a finite set, so it must be attained for some {£(k, j)}'j=1. Thus for all j A > « ^(M < iV(A, Therefore j j n(A,fi) = k = ^2e(k,j) < ^2N(X,Qj). □ The above lemma allows to prove Polya's conjecture for unions of cubes. Next, we have the "almost monotonicity" of \—> N(X, tt). Lemma. Consider open bounded sets £1 C £1 C Rd. Assume that there exists an extension operator E : H1^) —>■ H1^). Then 6.6. POLYA CONJECTURE 117 Proof. We use the min-max principle again 2 yUfc(Q) = inf sup . Mf!CZH1(Q)u€Mk JQ M dim Mk=k Actually we know that the infimum is attained at -^fc = Span{iti,uk} where {un}n C H1^) is an orthonormal basis of eigenfunctions of —AN(Q). Then we define Mfc = EMfc C H\n), dim Mfc = k. By the min-max principle on £1 /ife < sup ^--^ = sup A, 71 ■ ,GMfe k\V\ u^Mk k\Eu\ For every u 6 Mk, we can bound |V(Z^)|2+ / \Eu\2 = \\Eu\\2 ň <\\E\\2\\u\\2Hl{n) = \\E\\2([ |V^|2+ / M2) Jn Jn < \\E\\2((jik(ÍÍ) + l) í \u\2<\\E\\2((jik(il) + l) [\u\2. Jn Jn Hence, JaMEu)\ 2 fjLk(Q) < sup JSž; ,^J2y' < ||£||>fc(^) + 1). In particular, if N(\, tt) = k, then fj,k(tt) < A, and hence /ife(fi) < ||£||2(M^) + 1) < ||E||2(A + 1). Thus N(X,Q.) = k < N(\\E\\2(\ + l),n). □ In order to put the previous lemma in a good use, we need to control the norm of the extension operator in some simple cases. 118 CHAPTER 6. NEUMANN LAPLACIAN Lemma. Consider three cubes Q~ C Q C Q+ where Q+ = [—L+, L+]d, Q = [-L, L]d, Q_ = [—L~, L~] with L+ — L = L — L~. Take a closed set U C Q~. Then there exists an extension operator E : H\Q\U) ->■ H\Q+\U) with \\E\\2 < 2d. Is Q * t— \ u V ; \ (11UJ - Figure: Reflection x \-^-x. Proof. For every x g Q+\Q, we define the "reflection point" x g Q\Q~ by Xj, if g [—L, L], 2L-Xj, ifxj£[L,L+}, , Vj = l,2,...,d. -2L - Xj, if Xj g -L]. Note that each x has at most 2d — 1 preimages. Then for every / g HX(Q\U) we define the extension Ef g H^Q+XU) by (£/)(*) = f{x), XxeQ\U, f(x), if x g Q+\Q. Then it is straightforward to check that Ef g H (Q+\U) and ll^/ll HHQ+\U) < 2C H1(Q\U)- □ 6.6. POLY A CONJECTURE 119 Finally we can give Filonov's proof of Polya's conjecture for tiling domains. We know that Rd can be covered by disjoint sets {^n}^Li where each fln is a copy of fl. Let R = diam(Q). Take a big number L > 2R and denote Q = [-L,L]d, Q~ = [-(L-R),L-R]d, Q+ = [-(L + R), L + R]d. Let {Qjljgj be all copies inside Q~. Denote u = \JnjG Q~. Figure: C Q C Q By the sub-additivity of N(\, •) we have N(X, Q) ^j) + N(x> Q\u)- 120 CHAPTER 6. NEUMANN LAPLACIAN On the other hand, by the extension lemma, we can find an extension operator E : H\Q\U) ->■ H\Q+\U), \\E\\2 < 2d. Let {Qk}k£k be all copies inside Q+\U, namely Q\U C V = \JttkC Q+\U. Then E also defines an extension operator Ei : H\Q\U) ->■ H\mt(V)) C H\Q+\U), ||£i||2 < ||£||2 < 2d. Thus by the "almost monotonicity" and the sub-additivity of N(X, •), we have N(X, Q\U) < N(2d(X + 1), V) < E N(2d(X + 1), Qk). In conclusion, we already proved N(X, Q) oo we obtain m^<^N(x,n) (2ir)d ~ \ft\ 6.6. POLY A CONJECTURE 121 which is equivalent to k ' ' - (27t)d This completes the proof of Polya's conjecture for Neumann eigenvalues in tilling domains. □ Chapter 7 Many—body quantum systems We consider a system of N identical fermions in IRd. From first principles of quantum mechanics, the total energy of the system is described by a self-adjoint operator HN on the anti-symmetric space L2,(M>dN), which is a subspace of L2(M>dN) containing wave functions satisfying ^jvO^i) •••) xn •••) xji •••) xn) = ~^n(x1i •••) Xj,Xi,xN), Mi 7^ j, Vxj G Rd, or equivalently ^n{xa(i), ■-, xa{N)) = sign(cr)^Ar(xi,xN), Ma G SN where Sn is the permutation group of {1,2, ...,N}. A typical many-body Schrodinger operator has the form N i=l l • • • 8> 1 <8> ^/i^ (8)1 (8) • • • (8) 1, z-th variable and similarly for the two-body interaction Wiy The ground state energy is EN= inf (V,HNV). Il*lll,2=l 122 7.1. SLATER DETERMINANTS 123 If a ground state exists, then it solves the Schrodinger equation HN^ = EN^. 7.1 Slater determinants Definition (Slater determinants). For any functions {ui)f=1 in L2(M.d), define (ill A u2 A ... A uN)(x1, ...,xN) = —1= E sign(o-)iii(xCT(i))...iiAr(xCT(Ar)) VNl Clearly it is an anti-symmetric function in L2(M.dN J= E $lSa((T)U {Nl)-1/2sign{t) E ujtw{x1)...ujtw{x1s = (M)-1 E siSn(CT)sign(T)(^Mi)'^i))---^Miv)'^(iv)) (M) 1 J] sign(CT)sign(T)^(i)jV(i)-^(iv)jV(iv) 124 CHAPTER 7. MANY-BODY QUANTUM SYSTEMS 1, if ...,iN) = (ji,jN), 0, otherwise. It remains to prove that Ll(RdN) = Span {uH A u%2 A • • • A u%N | i±,..., iN G N, i\ < i2 < ■ ■ ■ < in}-Step 2. We prove that if tti, fl2 are two measure spaces, then L2 (fix x Q2) = L2^) (8) L2(Q2) := Span {n |n G L2(^i), v G L2(Q2)} where we used the usual notation of tensor product (u <8> y) = u(x)v(y). More precisely, we prove that if {"Ui}ieN is an orthonormal basis for L2 (Qi) and {"Ui}ieN for L2 (O2), then {ui C?> vJ}i^fij is an orthonormal basis for L2 (tti x Q2). {-Uj 8) gN are orthonormal functions in L2(£li X £l2) as (itj <8> ue 8) wfc) = ue)(vj,vk) = 5u5jk. {-Uj 8) "Ujj-j jeN is complete: Assume that / G L2(fli x Q2) and / _L u% (g> v3 for all i,j, then by Fubini's theorem x 7 J Jn1xn2 / f(x,y)vj(y)dfj,2(y)dfj,1(x). V-v-' Because {"Ui}ieN is an orthonormal basis for L2^^, we must have g3 = 0 in L2^!), namely for a.e. x G f{?,y)vj(y) = 0. Since this holds for j G N and {f^} -gN is an orthonormal basis for L2(fL), we find that for a.e. x G Sli, for a.e. y G Cl2, f(x,y) = 0. Thus / = 0 in L2(tti x Q2). 7.1. SLATER DETERMINANTS Using the above result and by induction, we find that L2 (WiN) = L2 (Rd) L2(Rd) • • • ulN | ix, ..., iN G N} is an orthonormal basis for L2(RdN). Step 3. We define the operator PN on L2 (RdN) by (PnVn)(xi, ■■•,xn) = j^^2 sign(o-)*w ■ ■ .,x*(n)) , V* N G L2 (Rd) . Then Pn is a projection as {Pn)2 = Pn' (Pn)2^n (Xl,..., xn) = PNJfi Yl sign(Cr)*Af (:Ml), • • ■ , ^a(iV)) = ' •••<£> ulN) = 0. Also, PN(ula(1) (8) • • • (8) nl(j(iV)) = sign(o-)PAr(itil (8) • • • ulN). Hence, L2(lRdAr) is equal to Span {PN(un ®---®uiff)\i1,...,iN G N} = Span \un A -ul2 A • • • A -uliV | zl5.. ., iN G N, i\ < i2 < • • • < In} ■ Thus the Slater determinants form an orthonormal basis for L2,(M>dN). □ 7.2 Reduced density matrices For many applications, the wave functions in L2(M.dN) have too many variables for practical computations. Therefore, it is often useful to consider its reduced density matrices which are simpler to analyze. / Definition. Let be a normalized wave function in L2a(RdN). The one—body density matrix 7^1') of is a trace class operator on L2(Rd) with kernel ^l(x,y) = N / ^N(x,X2,...,xN)^N(y,x2,...,xN)dx2...dxN. JWLd(N-l) 7.2. REDUCED DENSITY MATRICES 127 Thus 7^L') > 0 and Tr^f^ = N. The "diagonal part" 0/7^ is the one-body density p^N(x) = 7^(x,x) = N J \^N(x,x2, ...,xN)\2dx2...dx N (N-l) which satisfies p^N > 0, JRd p^N = N. In application, the one-body density matrix 7^ is sufficient to encode the expectation against every one-body observable: N '^N,^2h^N)=TT(h1iH). 1=1 Here as usual Tr(h^ ) = ^((7^ )1/2h(^] )1/2) makes sense when h is bounded from below on L2(IRd). In particular, if V is a multiplication operator on L2(lRd), then N „ ^N,yZV(Xl)^N) =Ti(V^l) = / V(x)pyN(x)dx. i=i 1 d Historically, the formalism of density matrices was introduced by John von Neumann in 1927. In his development of quantum statistical mechanics, the name "density matrix" is related to its analogue in classical statistical mechanics, namely a probability measure on the phase-space lRd x lRd. In this general setting, a mixed quantum state of N fermions is a trace class operator IV on L2(lRdAr) with TN>0, Tr r^v = 1 • Its one-body density matrix is obtained by taking the partial trace of all but 1 particle Tft =NTr2_+NTN. Thus T$ is a trace class operator on L2(lRd) with r$ > 0, Tr r$ = N. In terms of kernels, we can write T^\x,y) = N / TN(x,x2, ...,xN;y,x2, xN)dx2...dxN 128 CHAPTER 7. MANY-BODY QUANTUM SYSTEMS which is conceptually related to the the marginal distribution in probability theory. In particular, if TN is a pure state, namely TN = \^n){^n\ with a normalized function VN G L2a(RdN), then TN(x1,xN; yi, yN) = ^(^1, Zjv^jvG/i, ■•-,%) and T$ boils down to the operator 7^L') we defined before. A key consequence of the anti-symmetry assumption is Theorem (Pauli's exclusion principle). For every normalized wave function ^N G L2a(RdN), we have 0 < 7*1 < 1 on L2(Rd). Remarks: • Without the anti-symmetry assumption, 7^ may have an eigenvalue as large as N. In fact, if ^jv(^i) •••) xn) = u®N(xi,xjv) = u(xi)...u(xn) with a normalized function u G L2(lRd), then 7^ = N\u)(u\. • The physical interpretation of Pauli's exclusion principle is that "two quantum particles cannot occupy the same quantum state". A less precise version of this principle can be seen easily from the anti-symmetry assumption: if Xj = Xj for i ^ j, then ^jv(^i) •••) Xi, •••) xj, •••) xn) = —^n(xi, Xj,Xi,xn) = 0. However, the operator inequality 0 < 7^ < 1 is much deeper than the fact that the wave functions vanish on the diagonal set. It is easy to verify Pauli's exclusion principle for Slater determinants - - I .\(Mvi>r. Let {uj}f=1 be orthonormal functions in L2(R.d) and consider the Slater determinant = «i A «2 A ... A un. Prove that the one-body density matrix of 7.2. REDUCED DENSITY MATRICES 129 is N 7?J = £M<- i=l Proof of Pauli's exclusion principle in the general case. We want to prove that for every normalized function h6L2 , then By the definition of the one-body density matrix 7^, we can write Thus we need to prove that Take an orthonormal basis {u{\°cL1 for L2(lRd) such that -ui = u. We claim that A = £ litjj A • • • A uiN){uh A • • • A ulN\ l=lldN). Indeed, for every 1 < i\ < i2 < ... < in we have N ^ 1 on i^„\ii?dN^ Aun A • • • A ulN = ^T(Pu)xj -^=sign((r)ui(r(1)(a;i) • • • ula(N)(xN) = N 3=1 a€SN N 1(1 G {ii,..., ijv}) £ ^=sign((j)iiMl)(xi)---^ (iV) \XNJ -Ujj a • • • a ulN l = n, 0, otherwise. (j) \Xi) ' ' ' Uia(N) \XN) 130 CHAPTER 7. MANY-BODY QUANTUM SYSTEMS This completes the proof of Pauli's exclusion principle. □ The following result of Coleman (1963) tells us that the condition 0 < 7 < 1 in Pauli's exclusion principle is optimal. -\ Theorem (Admissible one-body density matrices). Let 7 be a trace class operator on L2(Rd) such that 0<7<1 onL2(Rd), Tr 7 = iV G N. Then there exists a mixed state TN on L2(M.dN), namely a non-negative operator on L2(M.dN) with TrT^v = 1, such that its one-body density matrix is TN^ = 7. Remarks: If 7 is a projection, namely 7 = 72, then Tn can be chosen to be a pure state Tn = \^n){^nI and is simply a Slater determinant. In general, it might be not possible to choose Tat to be a pure state (see an exercise below). Exercise. Let 7 be a trace class operator on L2(M.d) such that 0<7<1 onL2(Rd), Tr 7 = iV G N. Assume further that 7 has N — 1 eigenvalues equal to 1, but 7 is not a projection. Prove that there exists no normalized function ^at G L2(M.dN) such that = 7. Proof of Coleman's theorem. We claim that 7 can be written as a convex combination of rank-iV projections, namely 00 00 7 = ckjk, ck > 0, ck = 1, 7fc is a rank-iV projection for all k. fc=i fc=i Then any jk is the one-body density matrix of a Slater determinant ^Nik G L2(M.dN), and we can simply take 00 TN = J2ck\VNtk){V! 'N,k\- k=l 7.2. REDUCED DENSITY MATRICES 131 We construct the sequences {ck} and {'Jk} by induction. For k = 1, by Spectral Theorem we can write 00 7 = ^v ^n(l)\un){un\ 71=1 where {un}^^ is an orthonormal family in L2(Rd) and 00 Ai(7) > A2(7) > ... > 0, £ A„(7) = N. n=l If Atv+i(7) = 0, then 7 is a rank-iV projection and we can stop. Otherwise, we take £1 = min{AAr(7), 1 - AAr+i(7)} G (0,1) and write N 7 = £171 + (1 - £1)71, 71 = ^2\un)(un\, 0<7i^-i- K)(Un + V i-K)(^n|- z—' 1 — £1 z—' 1 — £1 n=l 1 n=7V+l 1 The choice of e\ ensures that 0 < 71 < 1. Of course, we take c\ = e\. For k = 2, we can repeat the above argument with 7 replaced by 71. More precisely, if 71 is a rank-iV projection, then we can stop. Otherwise, we can write 71 = £272 + (1 — £2)72, 72 rank-iV projection, 0 < 72 < 1, Tr72 = N and £2 = min{Ajv(7i), 1 — Ajv+i(7i)} G (0,1). Thus 7 = £171 + (1 - £1)71 = £i7i + (1 - £1)^272 + (1 - £i)(l - £2)72-Hence, we take c2 = (1 — £i)£2-For every k > 3, by induction we have 7 = ci7i + c272 + ••• + cfc_i7fc_i + (1 - £i)...(l - £fc-i)7fc-i. 132 CHAPTER 7. MANY-BODY QUANTUM SYSTEMS If 7fc_i is a rank-A projection, then we can stop. Otherwise, we can write 7fc_i = £fc7fc + (1 - £fc)7fc, 7fc rank-N projection, 0 < 7fc < 1, Tr7fc = A and £fc = minlA^TVi), 1 - AAr+i(7fc_i)} G (0,1). Thus 7 = ci7i + ••• + cfc-i7fc-i + (1 - - £fc-i)£fc7fc + (1 - £i)-(l - £fc-i)(l - £fc)7fc fc = ci7i + ... + cfc_i7fc_i + cfc7fc + JJ(1 - £j)7fc i=i with fc-i cfe = (1 - £i)...(l - £fc-i)£fc = £fc JJ(1 - £i). i=l Conclusion: In order to conclude oo 7 = E Cfe7fe fc=l it remains to show that oo oo £c, = l « JJ(l-efe) = 0 fc=i fc=i Assume by contradiction that Ecfc = l-5<1 « JJ(1 - £fc) = 5 > 0. fc=i fc=i Then we can write 7 = E Cfe7fe + ^7o0' 0 ^ Too < 1, Tr 7oo = A. k=l From the induction formula M M 7 = ECfe7fe + II^1 ~~ £k)lM, 0 < 7m < 1, Tr7M = A fc=i fc=i we find that 7m —>■ 7oo strongly in trace class as M —> oo. Consequently, £M+i = min{AAr(7M), 1 - Ajv+i(7m)} ->■ min{Ajv(7oo), 1 - Aat+i(7oo)} > 0. 7.3. IDEAL FERMI GAS 133 _ N i=i Here in the latter inequality, we have used that 1 N^ „ 1 Aat(7oo) > 0, \N+i{jco) < ---- Y A*(7co) < ---- Tr i=i But the fact that liniM->oo £m > 0 just contradicts the assumption oo n(i-£fe) >°- k=l This completes the proof. □ Remark: In general, we can also define higher reduced density matrices. For example, if is a normalized wave function in L2(IRdAr), then the two—body density matrix 7^ is a trace class operator on L2(IR2d) with kernel N{N-1) ^l(x1,x2;y1,y2) = ^' v^'—— / ^N(xt, x2, z3,zN)^N(y1,y2, z3,zN)dz3...dzN. ^ Jm.d(N-2) Then for every two-body operator W on L2(IR2d), we can write 1 1, the Hamiltonian N S,dN\ i=i 134 CHAPTER 7. MANY-BODY QUANTUM SYSTEMS is bounded from below with the core domain D(h)®- ■ -^Dih), and hence it can be defined as a self-adjoint operator by Friedrichs' method. Moreover, its ground state energy is n EN = y^fj,j(h) 1=1 where /ii(h) < /12(h) < ... are the min-max values of h. Proof. Using h > /ii(h), we have the obvious lower bound n HN = ^2hi>Nti1(h). i=i Thus Hn is bounded from below and hence it can be defined as a self-adjoint operator by Friedrichs' method. It remains to compute the ground state energy En of //TV-Lower bound. For every normalized wave function ^at G L2a(JsLdN) we can write n 1=1 Since 0 < 7^ < 1 (by Pauli's exclusion principle) and Tr 7^ = N, we obtain 00 00 Tr(fc7£) ^ inf { E hu") I K}~=i ONF, 0<^<1,J> = ^} n=l n=l n n = inf { ^k, hun) I K}™=i ONF} = n=l i=l thanks to the min-max principle. Upper bound. Consider the Slater determinant ^n = u\ A • • • A un- with orthonormal functions {ui)f=1 in L2(M.d) (we can take {ut}f=1 C D(h)). Then using n 71=1 7.3. IDEAL FERMI GAS 135 we have n {*N, HNVN) = Tr(h^N) = ^K, hun). n=l Thus n n EN < inf { ^K, hun) | {uX=i ONF} = n=l i=l □ Let us consider an example of the hydrogen-like atom. Theorem. For every iVeN, consider the Hamiltonian N A HN = ^(-AXt-vn) onL2a i=i Then is a self-adjoint operator with the quadratic form domain H^M. ) and its ground state energy satisfies By the general theory of the ideal gas, we know that n EN = ^2^(-A-N\x\-1) i=i where A — A|x|_1) is the z-th min-max value of the Schrodinger operator —A — A|x|_1 on L2(IR3). Actually, the spectrum of —A — A|x|_1 is known completely: it has negative eigenvalues A2 --— with multiplicity k2, with k = 1,2,... 4£r Hence, if we can write A = l2 + 22 + ... + M2 + M', 0■ Nx^3x we can write n n n En = 5>.(-A - N\x\-V) = Y^(-N2/3A - N^x]-1) = iV2'3 J^-A - N^x]-1). i=l i=l i=l Thus we need to show that n EN := 5>("A " N^lxl-1) = -N5'3^ + O(l)*_>oo). ,31/3 i=i Lower bound. For every constant a > 0, we can write n n EN ■= X^(-A - N^lx]-1) = J^/ii(-A + N2'3(a - M"1)) - aN5^3 i=i i=i > Tr(—A + N2/3(a - {x]'1)). - aN5/3. By Weyl's law, Tr(—A + N2'3{a - \x\-v))_ = -L?3 [ |iV2/3(a - {x^U^dx + o((N2/3)5/2)N^ = -N^fLf3 f |(a_|x|-l)_|5/2dx + o(1)Ar^co where If, = / |(|2**|» - l).|d* =(^.JM) = " ■ ^ = 1,3 JR3 m 1 ; 1 Vd + 2 (27r)d/|d=3 5 (2tt)3 15tt2 7.3. IDEAL FERMI GAS 137 and / \(a-\x\-1)_f2dx = ^ f\r-1-lf2r2dr=A7r ^ Jm? v a Jo 16' Thus in summary, En > "^5/3 ■ % ■ ^ + a + o(l)^J = -iV5/3 (-^ + a + o(l)^J Vlo7rz va lo / \12va ' We can optimize over a > 0, namely choose a > 0 such that 1 1 /I 1 \1/3 , x 9n -7= =-T= = a=[-=--= -a) = 24 "2/3. 24^a 24^a V247a 24^a / v ; Hence, EN > -N5/3 + 0(1)^) = -iV5/3 (— + 0(1)^00) • Upper bound. We need to show that with V(x) = a — \x\ x, a = (24) 2/3 and A = N2^3 N „ J>(-A + AF(x)) < -A^fc / \V-1=1 V ^3 \5/2 + o(l) By the min-max principle, at E^(-A + XV(x)) < Tr((-A + XV(x))-f) i=i for any trace class operator 7 on L2(1R3) satisfying 0<7<1, Tr7 = iV. We will construct a trial operator 7 using the coherent state method. Take a radial function 0 0 7.3. IDEAL FERMI GAS strongly in L3~(IR3). Combining with |V1|3/2 G L1 n L2~, we get \V.\3/2(V *G2n — V) = \V.\3/2(f - f * G2n) -)■ 0 strongly in L1(1R3). Thus we conclude that limsupA-5/2Tr((-A + \V{x))1) < -Lf;i f \V_\5/ A—>oo J This completes the proof of the upper bound. Chapter 8 Thomas—Fermi theory 8.1 Density functional theory In density functional theory, instead of considering a complicated wave function £ L2,(RdN) one simply looks at its one-body density N-l p^N(x) = N / \^n(x,X2, ...,xn)\ dx2...dxN, J{WLd which satisfies the simple constraints P^N{X) _t 0, / p^N(x)dx = N. Theorem (Representability). Let d > 1 and 0 < p G L1(Rd), JRd p = N G N. Then there exists a normalized wave function G -^a(IRdAr) such that p^N = p. We can choose $s G L^(M.dN) to be a Slater determinant. Moreover, we can choose $s G H~i(RdN) if and only if Jp G H1^). Proof. Step 1. For every 0 < p G L1(Rd), JRd p = N G N, we can take = ui A ii2 A ... A un, Uk = 2-Kik j\x) /N 140 8.1. DENSITY FUNCTIONAL THEORY 141 with a function / : IRd —> R. Then all of uk are normalized in L2(IRd) and N Y \uk(x)\2 = P(x)- k=l It remains to choose / such that {uk}^=1 are orthogonal. We use an idea of Harriman and Lieb (1981). Using the notation x = (xx,y) G M. x we define /(x) = /(x1) = l f ( / p(t,y)dy)dt. Then /'(x1) = — / p(x1,y)dy. Thus /' G i1(IR), hence / is at least continuous. Moreover, / is increasing (as p > 0)and lim f{xi) = 0, lim fix1) = 1. X1 —> — oo x1—>oo Moreover, for every k ^ £, we have / ^^(x)dx = / -Me^-^)dx = / ( f P^le^-^1)dy)dx1 = I /'(^le^'-^W1 = I'e2^-k>ds = 0. Jr Jo Thus {tifclj^L-L are orthonormal, and hence is a Slater determinant with TV k=l Step 2. If yfp G i/1^), then Vuk(x) = (Viyfpjx]) + yfpjx)2mkf\x1)) ^_ G L2(Md). Here V(-\/p(x)) £ L2(Rd) by the assumption ^Jp G i/1(lRd). For the second term, /'(a;1)\J p{x) G L2(lRd) because G H1^) G Lp(Md) for some p > 2 by Sobolev inequality and /'(x1) G 142 CHAPTER 8. THOMAS-FERMI THEORY L9(R) for all 1 < q < oo. The latter fact can be seen from l r „ , 1 ,, l /"(x1) = - / dXlP(x1,y)dy = - / dXl(Vp{x\y))2dy = T7 / y/p(x\y)dXl y/p(x\y)dy G LX(M) and Sobolev inequality W1'1^) C Lq(R) for all 1 < q < oo. Thus we conclude that if ^fp G i/1(]Rd), then all -u^ belong to i/1(lRd), and hence the Slater determinant constructed above belongs to H^(RdN). Step 3. If ^N G H^(RdN) (not necessarily a Slater determinant), then ^fp^ G H1^) because we have the Hoffmann-Ostenhof2 inequality n ^N,y2(-AXi)^N)> ivvp^i2 i=i ^d Using the one-body density matrix 7^L') , this inequality can be written as Tr(-A7« ) > f |V^ ■JRd Using the spectral decomposition 2 N I ' £i/n>l (here /n's not necessarily normalized), the above inequality can be written as 1/2 2 E/jvm2> /Jv(Ei/-ia): n>ljRd jRdl ™>1 ' For the sum of two functions, this follows directly from the diamagnetic inequality |V|y?(x)| | < |Vty?(x)| for every ip G i/1(lRd). For the general sum, we can do induction. □ The idea of describing a quantum state using only its one-body density goes back to Thomas and Fermi in 1927. It was conceptually pushed forward by a variational principle of Hohenberg and Kohn in 1964. Here we will follow the approach by Levy (1979) and Lieb (1983). In general, given any Hamiltonian Hn on L2l(RdN), the ground state energy can be 8.1. DENSITY FUNCTIONAL THEORY 143 rewritten as EN = inf (^N,HN^N)= inf inf (^N,HN^N). \\^N\\L2(RdN}=l P>0 \\^N\\L2(RdN}=l JRdP-N p-aN=P This motivates the definition of the Levy—Lieb density functional CN(p)= inf {Vn,HnVn), Vp>0, / p = N. H*^lli2(RdiV)=l jRd p-aN=P Thus the ground state problem of Hn becomes EN = inf CN(p). p>0 lud P=N This looks simple, but of course the complication of the many-body problem is now hidden in the determination of CN. In principle, computing CN is very hard. However, we may try to develop approximations which capture some properties of Cn when N —> oo. Consider a typical Hamiltonian HN on L2,(m>dn) of the form n hn = ^2[- h2AXi + V(Xl)^ + A £ w(Xl - xj). 1 = 1 1 R is an external potential and w : lRd —> R, w(x) = w(—x), is an interaction potential. The parameter h > 0 plays the role of Planck's constant and A > 0 corresponds to the strength of the interaction. For the external potential, we have the exact formula / N r 'vN, y2v{xi)^N) = / V{x)p*N{x)dx. 1=1 For the kinetic and interaction terms, there are no exact expression in terms of p^N. However, the semiclassical approximation suggests that 144 CHAPTER 8. THOMAS-FERMI THEORY while the mean—field approximation tell us Putting all this together, we arrive at the Thomas-Fermi approximation A (vn,HnVn) ~ Kfh2 P^/d + / VpyN + - PvN(x)pyN(y)w(x -y)dxdy. N ' JRd JRd ^ J JRdxRd In particular, this suggests that CN(p)*K?h2 [ p1+2ld+f Vp+^-ff p(x)p(y)w(x-y)dxdy. We will justify this approximation in the next section (following my joint work with Nina Gottschling (2018)). As we will see, the Thomas-Fermi theory is correct to the leading order in the semiclassical mean-field regime h ~ N~1/d, A ~ N~\ N oo. This is the choice making all three terms on the Thomas-Fermi density functional comparable (all are of order N). Historically, the Thomas-Fermi approximation was proposed for the atomic Hamiltonian, when V and w are Coulomb potentials in R3. We will consider it in a more general context, and then pay a special attention to the Coulomb case at the end. 8.2 Convergence of the kinetic density functional Recall the semiclassical approximation N *W,E(-A,J^)-W PIT' „-_i ' -JRd d (2?r) 2 d + 2 iSil2^' i=i 1 1 It is convenient to introduce UN == I \^N(x,x2, ...,xN)\2dx2...dxN, / > 0, / / = 1. 8.2. CONVERGENCE OF THE KINETIC DENSITY FUNCTIONAL 145 Thus the above approximation becomes 1 N r We can justify this approximation as follows. Theorem. For all d > 1, the followings hold true when N —> oo. (i) (Lower bound) If the normalized wave functions ^N G L2(M.dN) satisfy that f^N — f weakly in L1+2/d(Rd), then 1 N r (ii) (Upper bound) For every 0 < / G Lx(Md) n L1+2/d(Md), / / = 1, £/iere exist Slater determinants ^at G L2(IRdAr) suc/i i/iai —>■ / strongly in L1(Rd)r]L1+2/d(Rd) and >»-p^(*-E-^*,) E -a a) = j^miTr [(-A - N2/du)^N] + £ ATl+2/d i=l By the Pauli's exclusion principle 0 < 7^ < 1 and Weyl's law on the sum of negative eigenvalues, we can estimate Tr {-A-N2/dU)f^ > Tr[-A - N2'dU}_ = -Lf4 / \N2'dU\1+d/2 + o{{N2'd)1+dl2) 146 CHAPTER 8. THOMAS-FERMI THEORY = N1+2/d(-Lfd [ U1+d/2 + o(l)N^c V ' had Moreover, using feN —^ / weakly in L1+2/d(IRd) we find that uf. Thus Optimizing over U (i.e. choosing U = const.f2^d) we conclude that Upper bound. We can follow the coherent state approach in the proof of Weyl's law to deduce the upper bound, but in this way it is not easy to keep the important constraint that ^tv are Slater determinants. In the following, we will follow a more direct approach, which is close to Weyl's original method and Thomas-Fermi heuristic argument. Step 1 (Slater determinants of Dirichlet Laplacian on a cube). Consider the Dirichlet Laplacian —A on Q = [0,L]d. Recall that that it has eigenvalues \irk/L\2, k G Nd, with eigenfunctions uk(x) = n i=i 2 firklxl LSm[ — k = (kl)f=1, x = {xTi=ie The ground state of the M-body kinetic operator ^ .=1(—Ax ) is the Slater determinant M made of the first M eigenfunctions {uk}. It is straightforward to see that when M —> oo, 1 / M \ 1 1+2/d Ml+2/d s M i=l Ml+2/d irk T and first M eigenfunctions 1 \uk\2 ->• strongly in 27 (Q), \/p G [1, oo). Step 2 (Slater determinants of step-function densities). Let 8.2. convergence of the kinetic density functional 147 0 < / g L1^) n l1+2/d{rd), f f = 1. Let {q} be a finite family of disjoint cubes, whose construction will be specified in the next step. In the following we only consider cubes q such that />0. Q We can find an integer number mqg (n J f)±[-i,i] such that Y,MQ = n [ f = n. Now for every q, consider the first Mq eigenfunctions {uj}1^ of the Dirichlet Laplacian —A on q. These functions can be trivially extended to zero outside q to become a function Q\mq r~ UlfTO>d\ in Since the cubes {q} are disjoint, the n functions UoOvh-Ji c #o are orthonormal in l2 ^ family. Then in the limit n d\ Let g l2 r>dN\ N c -LJa^ ) be the Slater determinant made of this orthogonal + oo, using the fact that M, Q n f>0 Q and the calculation in Step 1 for each cube, we get N i=i N / Jy-l+2/d 1 mq ,<3||2 Q i=l Mq +21 d E'IV Q 112 1+1 Id Q i=l M, Q n 1+2/d Q \q\2'd 1+2/d Kd Y,\Q\\Hw\h1+Vd-Kd E /-f1+2/d ^ Jensen'sinequality) 1 q 1^1 jq q jq Q 1 we can find a L1 Ll + 2/d Using this collection of cubes, for every N > 1 we can construct a Slater determinant G L2a(RdN) as in Step 2. Thus there exists iVfc > 0 such that for every N > Nk, +2/d + k-l and L1 Q By the triangle inequality, for every N > Nk, l1 Ll+2/d Ll + 2/d < 2k~\ Now we conclude using a standard diagonal argument. By induction in k, we can choose the above sequence Nk such that Nk+i > Nk. Since lim^oo Nk = oo, we can find liniTv-^oo kN = oo slowly such that N>NkN. Thus the Slater determinant = £ Sn constructed as above satisfies, as N —> oo, i=i and This completes the proof of the theorem. □ 8.2. CONVERGENCE OF THE KINETIC DENSITY FUNCTIONAL 149 Note that the above theorem is conceptually equivalent to Weyl's law for the sum of eigenvalues. For example, we can use this theorem to give another analysis for the hydrogen-like atom We want to show that the ground state energy of HN is By rescaling x \—> Nx^x, it is equivalent to prove that the Hamiltonian n i=i ' ' has the ground state energy EN = —(3)1/3/4 + o(1)jv-kx>- Another look at H^. Lower bound. Take an arbitrary normalized wave function ^at G L2jRdN) such that '^N,HNyN) =EN + 0(N~1). By the Lieb-Thirring inequality with a constant K > 0 independent of N. Moreover, using JR3 feN = 1 and Holder's inequality we find that __-dx< /' -Hdx+ [ -Hdx i^i -'M>i H J\x\ —C and faN is bounded in L5/3(1R3). Up to a subsequence, we can assume that 150 CHAPTER 8. THOMAS-FERMI THEORY UN / weakly in L5/3(IR3). Then / < liminf / UN = 1. Moreover, by the above theorem, we have liminf (tyN,HNVN) > [ f/3-[ ^-dx N^oo \ I JK3 JK3 \X\ where 3 \d + 2\B1\2/dJ\d=3 5(4tt/3)2/3 5y ' ' Exercise. Consider the Thomas-Fermi functional = -W)2/3. £TF(f) = Kf[ fW-f Mdx, Kf = % Jm? Jm? \x\ & Prove that the variational problem E = inf \£TF{f) | 0 < / G L1(1R3) n L5/3(M3), f f < l) has a unique minimizer f0. Moreover, JR3 f0 = l and E = -(3)1/3/4. This leads to the lower bound liminf Eat = lim inf ( VN, HNVN ) > -(3)1/3/4 for a subsequence as N —> oo. We then obtain the convergence for the whole sequence by a standard contradiction argument. Upper bound. Let f0 be the Thomas-Fermi minimizer from the above exercise, JR3 f0 = 1. By the above theorem, we can find Slater determinants tyN G L2(lRdAr) such that f^N —> f0 strongly in Z/QR3) n L5/3(IR3) and lim sup ^ < lim sup (tyN, HNVN) = Kf [ /05/3 - I ^^dx = -(3)1/3/4. □ 8.3. CONVERGENCE OF THE LEVY-LIEB FUNCTIONAL 151 It is conjectured that N Kf f PIT < (^,£(-A^) < Kf f p\+2Jd + f |V^ JRd N • •, ' JRd JRd *? I AT < 7nk(»».D-^)*») * IC'Sj^'d+N~ViL 1=1 where the lower bound holds for all d > 3 (Lieb-Thirring conjecture) and the upper bound holds for all d > 1. The upper bound was proved by March and Young in 1958 for d = 1, but their proof cannot be extended to higher dimensions. 8.3 Convergence of the Levy—Lieb functional Consider the Hamiltonian N HN = ^2(-h2AXi + V(x^+\ £ i=l l R belong to Lp(Rd) + Lq(Rd) with p,q £ [1 + d/2, oo). Moreover, w admits the decomposition w(x) = (Xr* Xr)(x)dp(r), Jo 152 CHAPTER 8. THOMAS-FERMI THEORY for a positive measure/i on (0, oo) and for a family of even functions 0 < Xr £ Lp(Rd)+Lq(Rd) with p,q £ [2 + d, oo). The decomposition on w is equivalent to w(k)= / \xr(k)\2dp(r). Jo So this essentially requires that w(k) > 0, plus some regularity. This holds for a large class of potentials, including Coulomb potentials. For example, in 1R3 we have the Fefferman-de la Llave formula IIC°° dr n = - (iBr*iBr)(x)--, Viei3\{o} fI tt J0 rb where tsr is the characteristic function of the ball 5(0, r) in 1R3. Exercise. Let d > 1 and let 1 bt be the characteristic function of the ball -8(0, r) in Prove that for every 0 < A < d, there exists a constant C\:d > 0 such that 1 „ /"~ , ,, , dr \ id \X A r°° dr Cx4Jo (Ur*^)^)^^, VxeRd\{0}. Theorem (Gamma convergence from Levy-Lieb to Thomas-Fermi functional). For all d > 1, when N —>■ oo, the Levy-Lieb functional £^ converges to the Thomas-Fermi functional £TF in the following sense: (i) (Lower bound) For every sequence 0 < fN £ L1(Rd) Pi L1+2/d(Rd) such that JRd /at = 1 and Jn ^ f weakly in L1+2^d(Rd), then limM£N(fN)>£TF(f). N=co (ii) (Upper bound) For every 0 < / £ Lx(lRd) n L1+2/d(Rd) such that JRdf = 1, there exists a sequence of Slater determinants £ L2(RdN) such that f^N = f^ —> f strongly in Lx(lRd) n L1+2/d(Rd), and llmsup£N(fN)<£TF(f). N=oo Proof. Lower bound. Consider a normalized wave function VN £ L2a(RdN) with 8.3. CONVERGENCE OF THE LEVY-LIEB FUNCTIONAL 153 In -± f weakly in L1+2/d{Rd). We have N 1 N N ]\ll+2/d *N, y -AXi*N)+ i=i N2 1 0 and z G M.d, by the Cauchy-Schwarz inequality we get ■AT \ l i=l N i=l N vN,^2Xr(Xi-z)vn) - (vN,^2xl(Xi-z)vn i=l i=l 1 = ^[N2(fN*Xr)2(z)-N(fN*X2r)(z)}+. 154 CHAPTER 8. THOMAS-FERMI THEORY Since fN^f weakly in Lr(Rd) for all 1 < r < 1 + 2/d, and Xr, xl £ Lp(Rd) + Lq(Rd) with p,q G [1 + d/2, oo), we find that lim (/w * Xt)(z) = (f * Xt)(z), N-*oo Urn = (/*X?)(*). Hence, for every r > 0 and z G lRd, liminf iV"2 ( i'AT, Xr-(^i - - ^)^AT > TV—»00 \ ' J I \ l limmf iV"2I [iV2(/Ar * Xr)2(z) _ * x2)(z)] = !(/ * Xr)\z). Therefore, by Fatou's lemma, liminf N 2 ( E w(xi-Xj)^N \ l Xr(xi ~ z)Xr{xj ~ z)^n ) N^°° J0 -V \ 1<^[ dr [ dzhf * Xr)2(z) = \ [[ f(x)f(y)w(x - y)dxdy. JO JRd Z Z J JRdxRd Here we have repeatedly use the decomposition of w. Thus in summary, liminf(*£l£A)>£TF Since ^at G L2a(RdN) can be chosen arbitrarily under the sole condition feN = f^, this leads the desired lower bond limmf£N(fN)>£TF(f). Upper bound. Let 0 < / G Lx(lRd) n L1+2/d(Rd) with fRd f = 1. Then by the convergence of the kinetic functional, there exist Slater determinants tyN G L2,(RdN) such that f^N = fN-> f strongly in Lx(lRd) n L1+2/d(Rd) and (**• E-a-< >*") -Kf L/1+2/"- Since fN ->■ / in Lr(Rd) for all r G [1,1 + 2/d] and 1/ G Lp(Md) + Lq(Rd) with p,q £ 8.3. CONVERGENCE OF THE LEVY-LIEB FUNCTIONAL 155 [1 + d/2, oo), we have lim N-1(*N,J2v(xi)*N)= lim / VfN= f Vf. Finally, for the interaction terms, since tyN is a Slater determinants and w is non-negative, an explicit computation shows that p*N(x)pyN(y) - h^N(x,y)\2 w(x-y)dxdy l■ / in L\Rd) n L1+2/d(Rd) and the assumption w G Lp(Rd) + Lq(Rd) imply that fN*w^f*w inL°°(Md) by Young's inequality. Hence, N~2 ( ^n, Y w(Xl ~ xj)^n I - \ I / fN{x)fN{y)w{x - y)dxdy \ l 0, i/iera /TF is unique. We have ^Theorem(Coiivei^^^ 8.4. CONVERGENCE OF GROUND STATE ENERGY AND GROUND STATES 157 ground state energy E^ of converges to the Thomas-Fermi energy: lim ^ = £TF. Moreover, if^N £ L2(R.dN) is a ground state for , or more generally an approximate ground state in the sense that lim - = h , N^oo N then hN fF weakly in L1+2/d(Rd) where fTF is the unique Thomas-Fermi minimizer satisfying JRd /TF < 1. It may happen that EN has no minimizer, and/or ETF has no minimizer satisfying JRd fTF = 1- Nevertheless, the convergence of the ground state energy is always valid. In fact, the convergence of the ground state energy is valid under a very general condition on w (including negative potentials, e.g. w{x) = —as proved by Fournais, Lewin and Solovej (FLS-2018). This result justifies the validity of Thomas-Fermi in the atomic case, which was first proved by Lieb and Simon (1973). In this case we have Coulomb potentials in 1R3 V(x) = — -—-, w(x) = -—-. In the litterature, the atomic Hamiltonian is often written in the form N Hfr = T,{-A*<-r\)+ £ onL2a(R™), y \ T* - / nf ■ _ nf ■ i=l 1 11 1 N~x^x, Hf}oin is unitarily equivalent to N H^ = yfN2/3A » \ y .^i- on L2(RiN) i=l 1 11 l0 H*iV||I,2(HdiV) = l J«d/=1 f*N=f which can be rewritten as N / >o w; thanks to the definition of the (rescaled) Levy-Lieb functional {^N, HN^N) „. )=1 ^-IIT II ^ 1 /V HWjv||I,2(HdiV) = l iV Recall also the following equivalent definition of the Thomas-Fermi energy (see an exercise above) ETF = inf J£TF(/) : 0 < / G L\Rd) n L1+2/d(Md), ^ / = 1J . For every 0 < / G Lx(Md) n L1+2/d(Rd) satisfying fRdf = 1, by the Gamma convergence (upper bound), we can find Slater determinants ^ £ ^(R^) such that feN = f^ —> f strongly in Lx(lRd) n L1+2/d(IRd), and limsup^ < lim sup SN(fN) < STF(f). Then optimizing over / we obtain hm sup —— < E . 8.4. CONVERGENCE OF GROUND STATE ENERGY AND GROUND STATES 159 Energy lower bound. For any normalized wave function ^at G L2(lR3Ar), using w > 0 and the Lieb-Thirring kinetic inequality we have n Hence, i=i for a constant Kd > 0. Since V G Lp(IRd) + Lq(Rd) with p,q £ [1 + d/2, oo) we have (^v, HNVN) Kd f i+2/d J*N Thus En/N is bounded from below. Moreover, if the wave function satisfies then /at := fvN is bounded in L1+2/d(IRd). Up to a subsequence, we can assume that /at —^ / in L1+2/d(Rd). Hence, by the Gamma-convergence (lower bound) we have limine = liffiinfMM> Uminf > £TF(f) > £TF. In the last inequality, we have used the variational definition of ETF. Note that the weak convergence fN^f implies that 0 < / G L1(lRd) n L1+2/d(Rd) and /oo had Since the limit ETF is unique, we can obtain the lower bound estimate for the whole sequence N —> oo. Thus in conclusion we obtain the convergence of the ground state energy lim ^ = £TF. Convergence of ground states. Let v&jv be an approximate ground state for Hn and let fN = UN. Let /TF be a minimizer for ETF with JRd fTF = 1. Then from the above proof of 160 CHAPTER 8. THOMAS-FERMI THEORY the lower bound, we have fN -± f weakly in L1+2/d(Rd) and r En y {^n,Hn^>n) cTFfr\ jpTF cTF/^tfx lim — = hm--- = hm £N(fN) =£ (f) = E =£ (f ). Since the TF functional is strictly convex, it has a unique minimizer. Thus fN —^ /TF weakly in L1+2/d(Rd). □ 8.5 Atomic Thomas—Fermi minimizer We take a closer look at the Thomas-Fermi functional for Coulomb potentials == 3-k I -fMte+iff mm**. 5 Jm? JR3 \x\ ^JJm?xr3 \x ~ y\ with a constant K > 0 (the physical constant will be |K = Kf, namely K = (67r2)2/3). From an exercise of the previous section, we know that the minimization problem ETF := inf J£TF(/) : 0 < / G L1(1R3) n L5/3(M3), J / < 1J . has a unique minimizer /TF since wik) = const.\k\~2 > 0. Theorem (Atomic Thomas-Fermi minimizer). The unique atomic TF minimizer f is radially symmetric, JR3 fTF = 1, and it solves the TF equation KfTF{x)2'3 = \x\~x - fTF * \x\~\ Vx G M3\{0} Moreover, fTF is the unconstrained minimizer, namely £(fTF) < £TFtf), vo < / G L\R3) n L5/3(M3). tf Proof. Step 1. Since / i—> £TF(f) is rotational invariant, the unique minimizer /TF must be radially symmetric. Let us prove that JR3 fTF = 1- Assume by contradiction that JR3 fTF < 1-Then for every 0 <

0 small we have /TF+^>0, / (/TF + ^)<1. /TF i = r f fy> niy< [ CM^^L. 8.5. ATOMIC THOMAS-FERMI MINIMIZER 161 Thus by the minimality of /TF we have £TF(/TF)<£TF(/TF + ^) for all t > 0 sufficiently small. Consequently o<^(^TF(/TF + ^)) =/ (^(xp-A + ^^n)^)^- at V / |t=o+ Jr3 V \x\ |x|/ Since this holds for all 0 <

0, for a.e. x G M3. On the other hand, by Newton's theorem fTF(y)dy = f _ \x~y\ Jr3 max{|x|, \y\} - jr3 |x, , , Thus KfTF(x)2/3 >l--fF*l->(l-[ fF) JL for a.e. x G R3, |x| |x| V JR3 ' \x\ which implies that fTF(x) > for a.e. x G R3 Ixl6'2 with a constant c0 > 0. However, the last inequality contradicts to the fact that JR3 /TF < oo. Thus we must have JR3 /TF = 1. Step 2. Now we derive the TF equation. We can proceed similarly as above, but now we have to choose the test functions

-fTF(x) for a.e. x G R3, [ ip < 0. JR3 Then as above, we have £(/TF)<£TF(/TF + ^), VtG[o,i], and hence o<^F(r+tV)) =[ (K/TF(x)2/3--L+/TF*r^)^(x)dx. dt\ / \t=o+ JR3 V \x\ \x\J 162 CHAPTER 8. THOMAS-FERMI THEORY By taking

o. \x\ \x\/ ( f g).fTF(x) V/ro3 / 0 < g G CC°°(M3) Then the above variational inequality reads o< / (KfW/3-A + /TF*n 9<1. dx where W(x) Thus we have proved that / (K/TF(x)2/3-A + /TF*A)^)dx + ^F / g JE? v \X\ \x\/ 7R3 W(x)g(x)dx := KfTF(x)2/3 - -L + fTF * -1 + ^TF. M/(x)y(x)dx > 0, V0 < g G C\ 00 flD>3\ <7<1. which implies that VU(x) > 0 for a.e. x G IR3. On the other hand, thanks to the definition of y^TF and the fact that JR3 /TF = 0, we obtain / wr=[ (Kr(x)^--^+r*-^)r+pTF = o. Ju? jK3 v \x\ \x\/ Since W(x) > 0 and fTF(x) > 0 for a.e. x G IR3, we conclude that when fTF(x) > 0, we have W(x) = 0 namely Krxxf _ _]_ — f^F * —— — jl On the other hand, if / ix) = 1, then using W(x) > 0 we get 8.5. ATOMIC THOMAS-FERMI MINIMIZER 163 These two formulas can be written in the compact form _1 _ /TF - _ ^TF We will prove later that p = 0, but that requires a preparation. Step 3. Now we prove that /TF is an unconstrained minimizer. For any m > 1 (not necessarily an integer) we can consider the variational problem ETF(m) : = inf \sTF(f) : 0 < / G LX(R3) n L5/3(R3), / f -fm(x) for ip<0 we also have 0<^(fTF(/m + ^)) and that implies , Kfm{x)2'3--\ + fm*-\)^x)Ax |t=0+ ./h3 V \X\ \X\ ' /RS U/m(x)2/3 - A + /m * ^ fm{x) > 0. Now we take = <7(z) - (if )/m(x), 0 < g G CC°°(M3), / <7<1 v /Tri / _/in>3 ' Jr3 /l which gives 0< / (K/m(x)2/3-f+ /m*f Jr3 /r? dx 164 CHAPTER 8. THOMAS-FERMI THEORY (Kfm(x)2/S - + fm * -^-)g(x)dx +/im I g = I Wm(x)g(x)dx V \X\ \X\/ _ha3 7ro3 where * T7 +^m- \x\ Wm(x) := Kfm(xf3 -r-f + fr, \x\ This implies that Wm(x) > 0 for a.e. x G IR3. Moreover, by the choice of jim we have / Wmfm = I (Kfm(x)2/3 - ly + fm * fm + /im I fm = 0. 7r3 Jm.3 v \x\ \x\/ Thus Wm(x) = 0 if fm(x) > 0. All this gives the TF equation KUx)2'3 = ii /m * || l^r It remains to deduce the bound JR3 fm < 1 from the TF equation. Recall that yum > 0 and fm is radial (hence, we can use Newton's theorem to simplify the convolution). Multiplying the TF equation with fm{x) we have the pointwise inequality 0 < Kfm(xf3 = ■ 1 fm * = -\x\ -1 ~ fm * < -\x\ ■ 1 f -\x\ ./IB3 I \x\ 1 \x\ fm (x) fm{x) fm(y) -dy fm(x), for a.e. x G Integrating against < i?)|x|fedx with k = 2,3,... we obtain X\ fm(x)dx > \x\ b > 0 and k = 2,3,... we have the elementary inequality ak + bk max{a, b} which is equivalent to hk k — 1 = afe-i + __>___. ^-i + bk-u a k bk a"'1 + k—>(k- l)bk a 8.5. ATOMIC THOMAS-FERMI MINIMIZER 165 The latter follows from the AM-GM inequality ap-i + ik-T)— >kbk-\ a Coming back to the TF problem, we have f i f f \x\k + \y\k J\x\ 0 large we find that J^ 0, and hence i > ^ / UvW J\y\ oo and R —> oo we conclude that 1 > / fm- Thus fm = fTF, for all m > 1. This means that /TF is an unconstrained minimizer for the TF functional. Step 4. Since /TF is an unconstrained minimizer for the TF functional, we have £(fLb) <£(tfLb), W>0. Consequently, d ~dt (Kr(x)^--^+r*-\.r\x) \ \x\ \x\ -[1 tf Thus the TF equation becomes KfTF(x)^ = _±__j?TF ^ _±_ = A-/TF*A- \x\ \x\ 166 CHAPTER 8. THOMAS-FERMI THEORY Here in the second identity we have used Newton's theorem x\ Jm?\x — y\ JM3 max{|x|, \y\} JR3 |x| \x From the TF equation, we find that /TF(x) is continuous away from 0, since both |x|_1 and the convolution /TF * ^ are continuous away from 0. Hence, the TF equation holds true for all 0. This completes the proof of the theorem. □ Remark: The function *TF(^) = A - /TF * A > ° is called the Thomass—Fermi potential. Note that A$TF(x) = 4tt/tf(x), Vx G M3\{0}. Hence, the TF equation can be written in the equivalent form A$TF(x) = 4ttK-3/2$tf(x)3/2. This nonlinear PDE is well-studied. We have the Sommerfeld asymptotic formula lim |x|4$TF(x) = ATF, ATF = |x|-^oo 3ir2 In fact, the following pointwise bound was proved by Solovej (2000). ATF\x\~4 > $TF(x) > ATF\x\~4 - C\x\-4~\ e = > 0. II - \ ) - II II , 2 Consequently, the TF minimizer /TF(x) decays as const.|x|-6 when |x| —> oo. Chapter 9 Hartree—Fock theory Recall that the ground state energy of a Hamiltonian HN on L\{KdN) is defined by EN= inf {V,HNV). H*llL2(RdiV)=l In the Hartree-Fock theory, one restricts the consideration to Slater determinants and consider EHF = inf {V,HNV). \I/ a Slater determinant Thus by the variational principle, we have the obvious upper bound EN < £HF For a comparison, the Thomas-Fermi energy is neither an upper bound nor a lower bound to the full quantum energy. As we will see, in many situations, it is possible to obtain a good lower bound for En — EUF, making the Hartree-Fock theory significantly more precise than the Thomas-Fermi theory. In the same spirit of the density functional theory, an important advantage of the Hartree-Fock theory is that the energy expectation can be expressed purely in terms of the one-body density matrix of the Slater determinants. For a typical Hamiltonian H~n on L2a(JsLdN) of the form N HN = ^2 ( ~~ h2AXi + V(xt)J + A Y w(xi-Xj) i=l l0, Vx,yeMd. Thus the Hartree-Fock energy can be rewritten as £HF = inf £HF(7) 0<7=72<1 Tij=N Here the condition 7 = 72 is to ensure that 7 is a projection. For some computation, it is more convenient to ignore this condition since the set {0 < 7 < l,Tr7 = N} is convex. Actually, it is possible to do that without losing anything, provided that the interaction potential is non-negative. r Theorem (Lieb's variational principle). Ifw>0, th en EUF= inf £HF(7). 0<7<1 Tr7=Af Here V is assumed to be "regular enough" such that —h2A + V is bounded from below. This result was first proved by Lieb (1981). In the following we represent a simplified proof of Bach (1994). Proof. By a density argument, it suffices to take the infimum on the right side only on finite-rank operators. We will prove that if 0 < 7 < 1, Tr7 = N and £HF(7) < oo, then there 169 exists a projection 0 £HF(7). To construct 7, let us write M M 7 = 22 ^i\ui){ui\, 0 < Aj < 1, 22 Al = iui}i=i orthonormal functions in L2(M.d). i=i i=i Then we have M 1 M £nYd) = 22x^ + -222x^ i=l ij = l where Al = {ui, (-h2A+V)ui), Bij = 3ft / / (\ui(x)\2\uj(y)\2-Ui(x)uj(y)uj(x)ui(y))w(x-y) > 0. Note that Bl3 = B}1 and Btl = 0. Define 77(7) := |{Aj : 0 < Aj < 1}|. We assume that 77.(7) > 2; otherwise 7 is already a projection. Then there are at least two eigenvalues A^, A^ in (0,1) and we can assume that M M Ak + 22 Ai-Blfc > At + 22 Ai-Ew- 1=1 1=1 Let S = min{Afc, 1 — A^} > 0, then either Afc — S = 0 or A^ + S = 1. Define 7 = ( 22 Xi\ui)(ui\) + (Afc ~ $)\uk)(uk\ + (^ + S)\ut){uA. We have 77.(7) 1- n(l) ~ 1 and M M £HF(7) - £:HF(7) = XkAk + AM/; + 22 AlAfe£?lfe + £ A*A^« + X^Bke 170 CHAPTER 9. HARTREE-FOCK THEORY M M ((Afe - 5)Ak + (A* + 5)At + A,(Afc - d)Blk + E X^ + s)Ba + (A* ~ 5)(A^ + M M s(Ak-Ae + ^2 XiBik - E *iBit) + > 0. i=i i=i Thus we can replace 7 by 7 without increasing the energy. Here 7 may be not yet a projection, but since 77.(7) _~ ^(7) — 1, we can iterate the procedure and eventually obtain a projection after finitely many steps. □ In some situation, people are also interested in the reduced Hartree—Fock energy £rHF = inf £rHF(7) 0<7<1 Tr7=iV where £rHF(7) :=Tr((-/*2A + V07) + ^ / . / p7(x)p7(y)w(x - y)dxdy. Thus here we keep only the direct interaction energy and ignore the exchange energy. Mathematically, the reduced Hartree-Fock theory is easier to analyze since, for example, if w > 0 then 7 1—>■ £rHF(7) is convex. If w > 0, then the exchange energy is non-negative, and hence EN < £HF < £rHF. In this chapter, we will see how good the Hartree-Fock and reduced Hartree-Fock approximations are. The main question is the lower bound for En- We will focus on the atomic case where N ^ = E(-^2/3A--T^)+^-1 E T^hn onL^). i=l 1 11 l f, then 1 lT, \ ^ 1 f f fWKv) n^°° \ i<~^ Clo > 1-44. In the following we will prove the Lieb-Oxford inequality with a worse constant, using the strategy of Lieb, Solovej and Yngvason based on the Fefferman-de la Llave formula 1 1 f00 dr n = ~ (tBr*tBr)(x)-, Viei3\{o} \x\ tt J0 rb and the Hardy—Littlewood maximal function. 172 CHAPTER 9. HARTREE-FOCK THEORY Definition (Maximal function). For every f £ L11oc(lRd), the maximal function Mf : Rd —> [0, oo] is defined by r>o \x>{x, r) \ 7B(x>r.) Note that 1/0*01 = lim Tjrf-TT / l/(y)|dy < Mf(x), a.e. x £ r-^u \&\x, r)\ 7B(x>r.) On the other hand, the following result is very helpful. Theorem (Strong-type estimate). For every 1 < p < oo, we have \Mf\\LP(ßd^ < Cp||/||iP with a constant Cp £ (0, oo) independent of f. The original Hardy-Littlewood maximal inequality gives a constant Cd,P- The fact that Cp can be chosen independently of the dimension is due to Stein. Note that the condition p > 1 is crucial. For p = 1, we only have a weak-type estimate (see the proof below). Let us postpone the proof of the strong-type estimate and provide Proof of the Lieb-Oxford inequality. Let f be a normalized wave function in L2(lR3Ar) and denote p = p^. Recall that by the Fefferman-de la Llave representation 1 _1 r°° dr \x~y\ TT J0 r5 dzlBr(x - z)lBr(y - z), we can write For every r > 0 and z £ R3, by the Cauchy-Schwarz inequality we get Br{xi - z)tBr{xj - z)^> ) . l - ~ 2 i=i / \ i=i = l[(P*^Br)2(z)-(p*tBr)(z)} + = \(P* ^Br)2(z) - 1 min {(p * lßr)(z), (p * lßr)2(z)}. Integrating over z and dr/r5 we obtain -C [ dz [ ^min{(p*lßr)(z),(p*lßr)2(z)}. Jr3 Jo r 1 J It remains to bound the error term. By the definition of the maximal function, we have (p*lBr)(z)= / p(y)dy<\B(z,r)\Mp(z) = Cr6Mp(z). JB(z,r) Hence, for every R > 0 we can bound f°° dr r "l fR dr f00 dr / -min (p*lBr)(z),(p*lBr)2(z) < / -(p*tBr)2(z)+ -(p*lBr)(z) Jo r 1 J Jo r Jr r fR dr r°° dr < C / %reMp(z)2 + / %r3Mp(z) Jo r Jr r < C{R2Mp(z)2 + R^Mpiz)). We can optimize the right side over R > 0. It is easy to see that the optimizer R satisfies R2Mp(z)2 ~ R^Mpiz) ~ (tfM^zYiR^M^z))2)1'* = Mp{z)A'\ Thus we obtain r°° dr r -i J -mm{(p*lBr)(z),(p*lBr)2(z)} Lemma (Vitali covering lemma). Let {BA-j be a family of balls in M.d such that supdiam(Sj) < oo. Then there exists a subfamily of disjoint balls {Bj}ji such that [JB^d \j5Bj. Here if Bj = B(xj,rA , then 5Bj = B(xj, 5tj). Remarks: The set J can be finite, countable or uncountable. The subset J' is always at most countable since the balls {Bj}jr are disjoint. The condition sup-gi7diam(2?7-) < oo is crucial. Without it, a counter example is Bj = B(0,j) with j = 1,2,... The constant 5 is not optimal. It can be replaced by 3 + e (and 3 if J is finite). °°8 A °OXQ Figure from Wikipedia: The balls {Bj}j€j (left) and {3Bj}j€jr (right) Proof. Here let us consider the simple version when J is finite (the infinite case is harder and left as an exercise). We choose J' by induction. 9.1. LIEB-OXFORD INEQ UALITY 175 • First, we take a ball Bn of largest radius and put ji G J'. • Second, we ignore all balls with intersecting with Bn. If there is nothing left, then we stop. Otherwise, among all balls disjoint with Bn, we take a ball Bn of largest radius and put j2 G J7. • Assume that we have chosen jl5jk G J'. If every ball intersect with Bn U ... U BJk, then we stop. Otherwise, among all balls disjoint with Bn U ... U BJk, we take a ball Bjk of largest radius and put jfc+1 G J'. This procedure must stop after finitely many times. Then the resulting balls {Bj}j/ = {Bn, ...,Bm} are clearly disjoint. Moreover, any ball Bl with i G J, must intersect with a ball Bj with j G J' such that the radius of B% is < the radius of Br Then by the triangle inequality, B% c 3Bj C |J 3Br Consequently, \JBiC\j3Bj. i£j j€J' □ Now we are ready to provide Proof of the Hardy-Littlewood maximal inequality. Step 1. We prove the weak-type estimate supAKM; > A}| < Cd||/||Li(R,), V/ G L\Rd). A>0 Assume Mf(x) > A for some x G Rd. Then by the definition of Mf(x), we can find a ball B(x, rx) such that ,R/ M [ \f(y)\dy > A « \ [ \f(y)\dy > \B(x,rx)\. By the Vitali covering lemma, from the collection {B(x,rx) : x G J} with J = {x : Mfix) > A} we can find a sub-collection of disjoint balls {B(x,rx) : x G J'} such that \jB{x,rx) C |J 5B{x,rx). 176 CHAPTER 9. HARTREE-FOCK THEORY Consequently, \{Mf> A}| < | [JB{x,rx)\ < | |J 5B{x,rx)\ < ^ \5B{x, rx)\ xGJ xGJ' xGJ' = 5d22\B(x,rx)\<^22 I \f(y)\dy<54 [ A ^ v JB(x,rx) A Jr° \f(y)\dy. This completes the proof of the weak-type estimate. Step 2. Now we prove the strong-type estimate. For every / G Lp(lRd) we use the layer-cake representation / Mf(x)pdx = pX^liMf > \}\d\. JRd JO If we simply insert the weak-type estimate |{M/>A}|<^||/||ii in the layer-cake representation, then we get oo since Xp~2 is not integrable. However, we can split |/| = |/|1(|/| > A/2) + |/|1(|/| < A/2) < g + A/2, g = |/|1(|/| > A/2) which implies that Mf A}| < \{Mg > A/2}| < j^WgWw) = \m\H\m\ > A/2)dy where we have applied the weak-type estimate for g. Inserting the latter bound in the layer-cake representation and using Fubini's theorem, we conclude that Mf{x)pdx < Cd,p / dAAp~2 / dy\f(y)\l(\f(y)\ > A/2) J0 Jwtd = Cd,p / dy\f(y)\ / dAA^l(|/(y)| > A/2) < Cd,p / dy\f(y)\p. JWLd JO JWLd This completes the proof of the strong-type estimate. □ 9.2. ERROR BOUND FOR ATOMIC REDUCED HARTREE-FOCK ENERGY 177 9.2 Error bound for atomic reduced Hartree—Fock energy Now we come back to the ground state energy EN of the atomic Hamiltonian N HN = Y{-N-2/3^-rh)+N~1 £ yrh-\ ™L^- i=l 1 l| l {vn,hnvn) +0(n~1). By the Lieb-Oxford inequality hn*n) > Tr((-iV-/3A + VhU ) + ±.ff P^x)p*fy)dxdy - cn^ [ pfN Jm.3 Jr3 \x — y\ 7R3 = ^HF(7«)-cW f^. JE? As in the proof of the validity of Thomas-Fermi theory, we know that {*n,hn*n) >n(k [ UN(xf/3dx- [ -^f^dx) >n(^ [ UN(x)5/3dx-c) v Jr3 Jr3 \x\ / \ 2 Jr3 / 178 CHAPTER 9. HARTREE-FOCK THEORY for a constant K > 0. Since (^N, HN^N) < EN + 0{N-r) < CN, we find that hN{xf3dx f rHF(7£) - CN^3 [ > ErHF _ CArl/3_ This completes the proof of the desired lower bound for EN. Thus in summary, £rHF > EN > £rHF - CN1'3. □ Next, we compare the reduced Hartree-Fock energy .ErHF with the Thomas-Fermi energy £TF := inf (Kf [ f'3 - [ f4*±dx + l- ft ^^dxdy). o 0. 3 ^ Theorem. We have ErHF = NETF + oJ^n^. j This result is not new as we already proved En = NETF + o(N)n^>00 and En = ElF[F + O^N1'3). Nevertheless, the proof below gives another approach to the validity of the Thomas-Fermi theory EN = NETF + o(N)N^oo- Proof. Lower bound. Take 0 < 7 < 1 on L2(IR3), Tr7 = iV 9.2. ERROR BOUND FOR ATOMIC REDUCED HARTREE-FOCK ENERGY and consider £rHF(7) = Tr((-iV-V3A _ N-i)7) + ±D(py,Py) where z Jm? Jm? \x ~ V\ Since the Coulomb potential is positive-type, namely D(f, /) > 0, we have D(Pl, Pl) = D(Pl - NfF, Pl - NfF) - N2D(fTF, fF) + 2ND(Pl, fF) > -N2D(fTF, fTF) + 2ND(p7, fTF) = -N2D(fF,fF) + NTT((fF*\xn1). Hence, ^rHF(7) > Tr((-iV-2/3A - Ixl"1)^ - ND(fTF, fTF) + Tr((/TF * H"1^ = Tr((-iV-2/3A - $TF)7) - ND(fTF, fTF) > Tr(-iV-2/3A - $TF)_ - ND(fTF, fTF). Optimizing over 7 we obtain the lower bound ErHF > Tr(_iV-2/3a _ $TF)_ _ jV£)(/TF) /TF)_ Note that ETF + D(fTFJTF) = Kf [ (fTFf3- [ ^-dx+ [[ fTF(x)fTF{y)dxdy Jm.3 Jwl3 \x\ JJm.3xwl3 \x V\ = Kf [ (fTFf3 - [ /TF$TF = -Lg3 / ($TF)5/2 where we have used the pointwise equality Kcl(/TF)5/3 _ /TF$TF + Lcl3($TF)5/2 = g Recall that by the definition of Kf and Lf3 we have Kfa5/3 + Lf3b5/2 > ab, Va, b > 0 180 CHAPTER 9. HARTREE-FOCK THEORY and the equality occurs when b = -Kfa2!3 3 3 which is exactly verified by the TF equation when a = fTF and b = $TF. Thus we have proved that ErHF _ ^TF > Tr(_iV-2/3a _ $TF}_ + f ($TF)5/2_ Jm? Since 0 < $TF G L5/2(1R3), by Weyl's law we have lim N-1 Tr(-iV2/3A - $TF)_ = lim k^'2 Tr(A - K$TF)_ = -Lf3 [ ($TF)5/2. Thus E^F-NETF>o(N)N^. Upper bound. As in the proof of Weyl's law, we choose the trial state, with n = N2^3, !■= I I \Fk,y){FKy\t(\2Ttk\2 - K<$>TF(y) <0)dkdy. Then 0 < 7 < 1 on L2(R3) by the resolution of identity and Tr7= f f 1(|2ttA;|2-K$TF(y) < 0)dkdy = L$3 f |/«$TF(y)|3/2 = NLtJ (\Kir{x)2'3f2 = N. Here we have used JR3 /TF = 1 and °'3V3 3 J (2tt)3V3 5V ; / (2tt)3 Thus proceeding as in the Weyl's law upper bound, we find that Tr((-A - K$TF)7) = —Lft3 J(K$TF)5/2 + o{k"I2) 9.2. ERROR BOUND FOR ATOMIC REDUCED HARTREE-FOCK ENERGY which is equivalent to Tr((-JV"2/3A - $TF)7) = .\7.V,. J'($TF)5/2 + o(N). M3 It remains to bound the direct term -D(p7 — NfTF,p7 — NfTF). From the definition 7= // \Fk,y){FkJdkdy. J J|27rfc|2-K$TF(y)<0 and Fkty(x) = e2nihxG(x — y), ||G||l2(r3)=ij we can compute explicitly Pi(x) = = \Fk^y(x)\2dkdy J 7|27rfc|2-K$TF(j/)<0 = \\ \G(x - y)\2dkdy = Lga / (k$tf (y)f2\G(x - y)\2dy = N f fTF(y)\G(x-y)\2dy = N(fF*G2)(x). Here we are going to choose G2 = G2N —> 5 slowly as —> oo. Therefore, ,/TF * G2N ->■ /TF strongly in L1(1R3) n L5/3(R3). Consequently, N-2D(Pl - A7TF, Pl - NfF) = D(fF * GN - fF, fF * GN - ,fTF) 0. Thus we conclude that for the above choice of 7, ElFFF < £rHF(7) = Tr((-iV-2/3A - \x\-^) + 1d(p7jp7) = Tr((-iV-2/3A - $TF(x))7) - y/*.Y/". iV/TF) + ±D(Pj - iV/TF, p7 - Nf = -NLfj j($TF)5/2 - ND(fTF, fTF) + o(JV) r3 = A^ETF + o(iV). This completes the proof of the upper bound. Actually we have the following deeper result 182 CHAPTER 9. HARTREE-FOCK THEORY ^ Theorem (Scott correction). We have ErHF = NETF + |jV2/3 + o(iV2/3)Ar^00. J Since En = EtUF + O^N1^3), the Scott correction holds for the full energy En as well. The proof of the Scott correction is significantly more complicated than the derivation of the Thomas-Fermi energy; below we will give an outline of the main ideas. Sketch of the proof. Lower bound. From the above analysis we already showed that ErHF > Tr(_AT-2/3A _ $TF}_ _ ND(fF^ /TF) = NETF + Tr(-iV"2/3A - $TF)_ + NLf3 f ($TF)5/2. Thus we need to prove the following correction to the semiclassical approximation Tr(-iV-2/3A _ $TF}_ = _jVLcl3 f ($TF)5/2 + 1^2/3 + ^2/3^ ' Jm.3 8 Actually the contribution |iV2/3 comes from the particles moving very close to the nucleus (of a distance 0(iV_2/3) which is much smaller than the semiclassical distance 0(1)). The contribution of these particles is comparable to the non-interacting case, namely the hydrogen atom. Thus the key ingredient of the proof is the following Theorem (Hydrogen comparison). We have Tr(-iV-2/3A _ $TF}_ + NL*3 f ($TF)5/2 Jm? -(Tt(-N-2/3A- l^l^ + l).) + NLf3 f dxl"1 - l)^/2dx 3 Here we replace the potential by — 1 to ensure that J^O^I-1 — l)+2dx < oo. The proof of this result is rather complicated as we need some advanced tools beyond the coherent states discussed in the course. See e.g. Solovej and Spitzer (2002). To get the desired conclusion, we use the exact calculation for hydrogen. Recall that the operator —h2A — on L2(1R3) has eigenvalues —l/(Ah2n2) with multiplicity n2. Hence, ■ft(-^-N-' + i).= £ (!+„»)=- +0(ft->). l Tr(_AT-2/3A _ $TF}_ + f ($TF)5/2 Upper bound. We can show that there exists an operator 0 < 7 < 1 on L2(IR3) such that Tr 7 = N and Tr((-AT-2/3A - $TF)7) = Tr(-iV2/3A - $TF)_ + o(iV2/3), N-^D{Pl - NfTF, Pl - NfTF) < o{N2'3). The choice of 7 is more complicated than just / / |Ffc,,)(Ffc,,|l(|27rA:|2-K$TF(y)<0)dA:dy since we have to do something more precise in the domain {|x| < 0(N2/3)}. Actually the construction of 7 follows from the proof of the hydrogen comparison we mentioned above. □ 9.3 Bach's correlation inequality In this section we discuss an improvement of the Lieb-Oxford inequality where the exchange term is taken into account. ^Theorern^^ach^^oiTCl^ 184 CHAPTER 9. HARTREE-FOCK THEORY ^ G L2a(R3N) and denote 7 = 7^ and 7t = 7 - 72- Then l<3 \xi~xj\ I 2 JK3 JR3 |x — y| This inequality was proved by Bach (1992) in his proof of the accuracy of the Hartree-Fock energy (we will come to this later). Note that by Pauli's exclusion principle, the truncated one-body density matrix 7t = 7 — 72 satisfies 0 < 7x < 7 < 1. Consequently, 0 < plT < p1. Moreover, if ^ is a Slater determinant, then 7 is a projection and hence 7x = 0; in this case we have the equality N i^lxi-Xj] / 2JR3jR3 \x-y\ In applications, when 7 is close to a projection, then 7x is close to 0 and the error term in Bach's inequality is much smaller than that in the Lieb-Oxford inequality (with the price that we have included the exchange energy on the right side). Recall that the Lieb-Oxford inequality was proved using the Fefferman-de la Llave decomposition and Hardy-Littlewood maximal function together with the following bound for A — N I E XiXj* ) > ^(Tr[X7])2 - \ Tr(A7) Actually, this inequality easily follows from the identity N N _ N i 6 L2(R3Ar) and denote 7 = 7^L') and 7t = 7 — 72 • Then we have U^XX^j > ±((Tr[X7])2 -Tr(X7X7)) - CTr(X7) min{l, (Tr^x])1/2}. Proof. We follow the representation of Graf and Solovej (1994). Let P,Q be operators on L2(1R3) such that 1 = P + Q, 1>P,Q>0. Then we can decompose N N 22XiXj = 22(P + QMPj + g,)-V,.V;;/'; + Qj)(Pi + Qi) = A, + A2 + A3 where N A1 = 22(PQ3 + QiPj + QlQj)XlXJ(PlQJ + Q,P3 + QlQJ), A2 = — (PjPjXlXjPiPj + 2PlPJXlXJPlQJ + 2()J\\ ,\ tPJ\ ). N A, = Y,{PiPjXiXiQ iQ j QiQ j x^Xj P^Pjj. i 0 since X > 0. For ^2, we take an arbitrary constant a > 0 and complete the square 1 2 1 a - 22(PXP + 2PXQ), =-a-^(PI(l + Q))! 2 > 0 2 which is equivalent to A2 = \ YJ(kPXP)l(PXP)3 + 2(PXP)i(PXQ)j + 2(QXP)i(PXP)j) - ~\a2 + \zZ (2a(x " QXQ^ ~ t1 + Q)*^2*(i + Q)) - 2 yiQXPUPXQ), 186 CHAPTER 9. HARTREE-FOCK THEORY For A3, we take an arbitrary constant /3 > 0 and complete the square 1 r1/2Y.^xp)*-P1/2(pxQ) '2 which is equivalent to A3 = ^2((PXQ)i(PXQ)j + (QXP)i(QXP)j) >-\Y ((pxQ)2 + (QXP)2 + (3QXP2XQ + (3~1PXQ2XP^ i Thus in summary, for every a > 0 and /3 > 0 N 1 1 ^ -2ft2 - 2(4+^+ rl) E^xpMpx^ i 0 Thus 2a(X - QXQ) - (1 + Q)XP2X(1 + Q) - (PXQ)2 - (QXP)2 - j3QXP2XQ - I3~1PXQ2XP > 2a(X - QXQ) - XPX - (1 + 2fi)QXP2XQ - I3~1(XP2X + PXQ2XP) > 2a(X - QXQ) - XPX - (1 + 2p1)\\XP2X\\QXQ - ^{\\PfX + ||Q||2PXP) 9.3. BACH'S CORRELATION INEQUALITY 187 > 2a[X - QXQ) - XPX - (1 + 2/3) Ti(XP)QXQ - ^~\X + PXP) = 2aX - XPX - {2a + (1 + 2/3) Ti{XP))QXQ - /3_1(X + PXP). For the two-body part we have Y,(QXP)i(PXQ), = Y(pxpUQxQ)o ~\(piQi ~ Q1p3)x1x3(p1q3 - QiPj) < J2(PXP)i(QXQ)j < (E^^EWM) ^ Tr(PXP)(^(QXQ),-). i+3 i 3 3 Here in the last estimate we have used the Pauli's exclusion principle. Exercise. Let A be a nonnegative trace class operator on L2(Rd). Prove that N YA--a2 i<3 1 N + 2 E {2aX - XPX - (2a + (5 + 3/3 + Z3"1) Tr(XP))QXQ - /3"X(X + PXP)) . i=i Therefore, for every normalized wave function \I> G L2(lR3Ar) with 7 = 7^L') we have \ i<3 I + i Tr [^2aX - XPX - (2a + (5 + 3/3 + /3"1) Tr(XP))QXQ - /3"X(X + PXP))7 Choosing a = Tr[X7J we obtain U^XtXjVS > i(Tr[X7])2 - ^Tr[XPX7] \ i<3 I - ^ ((2 Tr[X7] + (5 + 3/3 + /3"1) Tr[XP]) Tr[QXQ7] + /3"1 Tr[(X + PXP)7]) Now we choose two different projections P to conclude. 188 CHAPTER 9. HARTREE-FOCK THEORY Choosing P = 1, then Q = 0 and we have U^XtXjVS > -^(Tr[X7])2 - ITr[X7] -r'TrtXT]. Then we can take /3 —> oo to get n V^XiXj*) > l(Ti[X7])2 - ^Tr[X7]. i 1, then we get the desired conclusion XiX^\ > ^(Tr[X7])2 - 1 Tr[X7] min{l, (Tr[X7T])1/2}. Choosing P = 7, then Q = 1 — 7. Using Tr(QXQ7) = Tr((l - 7)X(1 - 7)7) < Tr(X(l - 7)7) = Tr(X7x) we have 1 V^XiXjV) > ^(Tr[X7])2 - ^Tr[X7X7] (7 + 3/3 + ß-1) Tr[X7] Tr[X7x] + 2/3"1 Tr[X7] + Tr(X7X7]) 2 >i(Tr[X7])2-iTr[X7X7] - i Tr[X7] ((7 + 3/3 + /3"1) Tr[X7x] + 2/3"1). Now we only need to consider when Tr[X7x] < 1. In this case, inf ((7 + 3/3 + /3"1) Tr[X7T] + 2/3"1) < C(Tr[X7x])1/2 (if Tr[X7x] = 0 it is obvious; otherwise we can take /3 = (Tr[X7x])_1^2). Thus U^XtXjVS > i(Tr[X7])2 - iTr[X7X7] -CTr[X7]Tr[X7T]1/2. 9.3. BACH'S CORRELATION INEQUALITY 189 In summary, in all cases we have U^XiXjV) > ^(Tr[X7])2 - ^Tr[X7X7] - CTr[X7] min {l, Tr^x]1/2}. This completes the proof of the lemma. □ Now we are ready to provide Proof of Bach's correlation inequality. Let ^ be a normalized wave function in L2(R3Ar) and denote 7 = ^\ 7t = 7 — 72- Thanks to the Fefferman-de la Llave representation 1 f°° dr IF dz^B(z,r){x)tB(z,r) iv) \x~y\ ^ Jo r we can write / N 1 \ 1 1"°° d f N \ % 0 and z G R3, applying the previous lemma to X = \b(z,t) (this multiplication operator is a projection) we have N ^(v,tB^t){xi)Mz,t){xj)y) > ^((T^1^)^)2 - Tr(lB(ZiI.)7lB(ZiI.)7)) -CTr(lB(ZiI.)7) min{l,(Tr[lB(ZiI.)7T])1/2}. Using the Fefferman-de la Llave representation again we obtain 1 f°° dr f 1 f°° dr f f f ~ / dz(Tr[:lls(z,r)7])2 = - / -r dz dxtB(z^(x)p (x) / dy\B(z^(y)p (y) k Jo r Jm.3 k Jo r Jwl* Jm.3 Jm.3 P^x)p^y)-dxdy. \x - y\ Similarly, using (^B(z,r)l^B(z,r)l)(x,x) = / dytB^r)(x)7(x, y)lB(z,r)(y)l(y, x) = I dytB^r)(x)tB{z^r)(y)|7(x, y) I 190 CHAPTER 9. HARTREE-FOCK THEORY we can write 1 I"00 dr f , „ r n 1 Tdr 7T ./0 r ./IB3 7T dzTr[lB(z>r.)7lB(z>r.)7] = - / — / dz / dx / dylB(z>r)(x)lB(z>r)(y)|7(x, y)| |2 \x - y Thus AT ^:=i;(*.w*.)w^)»)-5/ / ^'^'-'^■""dPly f-f \ / 2JR3 yR3 |x - y\ >-C — cbTr(lB(^)7)min{l,(Tr[lB(^)7T])1/2}. Jo r Jr3 We control this error term by the maximal functions similarly the proof of the Lieb-Oxford inequality: Tr(lB(ZlI.)7) = / dyPi(v) < Cr3Mp(z), Mp{z) := sup 1 / Pl{y)dy JB(z,r) r>0 \-D{Z, r)\ JB(z,r) and Tr(lB(ZiI.)7T) < Cr3MPT(z), MPT(z) = sup 1 / P7T(y)dy. Thus r°° dr r A>-C — dzMp(z) mm{l, r^M^z)1/2}. Jo r Jr3 For every z G IR3, for every R = R(z) > 0 we can bound °° dr r „,„„ , x1/0, /"^dr „,„„ , x1/0 r°dr min{l,r3/2MPT(z)1/2}< / ^r^M^z)1/2 Jo r Then optimizing over R > 0 we get M^z^R}'2 ~ iT1 - ((M^^)1/2/?1/2)2/?"1)173 = M^z)1/3 we get 00 Hr * min{l, r^M^z)1/2} < CM^z)1'3. 9.4. ERROR BOUND FOR ATOMIC HARTREE-FOCK ENERGY 191 Thus 1/3 A>-C f dzMp{z)MPT{z ■JB.3 >-c(f dzMp(zf3X/Zt ( f dzMPT(zf3) \Jm.3 / \Jwl3 / >-c( f dzp^zY'3] ( f dzp^zf'3] \Jm.3 / \Jm.3 / Here we have used Holder's and the Hardy-Littlewood maximal inequalities. This completes the proof of Bach's correlation inequality. □ 9.4 Error bound for atomic Hartree—Fock energy In this section we compare the ground state energy En of the atomic Hamiltonian N i=\ 1 11 l Tr(/i7) - ND(fTF, fTF) + 0(N1/3) > Tr(/i_7) - Tr(/i_) + ETF + o(N) and EN 0, define Pe = l(h<-e). Using 0 < 7x = 7(1 — 7) < 7 and 7r < 1 — 7 we have Tr(7T) = Tr((l - P£)lT) + Tr(P£7T) < Tr((l - Pe)7) + Tr(Pe(l - 7)) = Tr(7) - Tr(P£) + 2 Tr(Pe(l - 7)). 9.4. ERROR BOUND FOR ATOMIC HARTREE-FOCK ENERGY 193 Combining with P£ < —h_/e we obtain Tr(7x) < N - Tr(P£) + 2s"1 Tr(/i_(l - 7)). Recall the estimate Tr(/i_(l — 7)) = o(N) in Step 1. Moreover, for every e > 0, by Weyl's law for the number of negative eigenvalues we have Tr(P£) = Trl(/i + £ < 0) = Trl(-A - iV2/3($TF -e) < 0) = NLc0l3 [ [$TF - e]f + o(N)N^. ■Jr3 Thus TF -e\f. lim sup N'1 Tr(7x) < 1 - Lc0\3 [ [$' n^oo ' Jm.3 Sending e —> 0+ on the right side, we conclude that lim sup N'1 Tr(7x) < 1 - Lg3 / ($TF)3/2 = 0 □ Exercise. Let 0 < V G Ld/2(Md) n L1+d/2(Md). Prove t/iat Tr 1(-A + XV < 0) = L*d / (XV)d/2 + o(Ad/2)A^co i7m£; You can use Weyl's law for sum of eigenvalues and a Tauberian argument. Now we are ready to give Proof of the estimate En — EUF = o(iV1/3). We only need to consider the lower bound for En — EUF. Take \I> an approximate ground state for En such that (V,HNV) = En + 0(iV_1). Let 7 = 7^L') and 7t = 7 — 72- By Bach's correlation inequality we have {%HN*) > Tr((-iv-/3A _ N-i)7) + ±ff p^)p,(y)-h^y)\\d^ 2^ Jr3 Jr3 \x - y\ 3/4 / r n 1/4 V/i»3 / V/i»3 194 CHAPTER 9. HARTREE-FOCK THEORY ZE^-CN-tf [ pf)3/V / p kjr3 / V Jr3 ,4/3* 1/4 7T It remains to bound the error term on the right side. By Holder's and the Lieb-Thirring inequalities 1/2 / r \ 1/2 P. v Jr3 7 v Jr3 For the truncated one-body density, we use p7{,3 < p7^3 for the kinetic term and the bound Tt7t = o(N) in the previous lemma P'S < ( I <)1/2( / P^)1/2 < C{N^yi2o{NY'2 = o(N^). kJr3 ' kJr3 ' Thus in summary, N-Hf Pff\( pt':)1" R is periodic, even, bounded and of positive type V(x) = £ V(k)elk-X, V(k) > 0, \/k G Z3. fcez3 9.5. HARTREE-FOCK ENERGY OF A HOMOGENEOUS FERMI GAS 195 The ground state energy of Hn is EN= inf (V,HNV). H*ll£lCK3JV) The corresponding Hartree-Fock energy is £HF = inf fTr(-iV-2/3A7) + -L / / L(x)p7(y) - |7(x, y)\2} V(x - y)dxdy] . 0<7=72<1 \ 2N Tr7=7V T3 jt3 Without the interaction (i.e. V = 0), the Hartree-Fock minimizer is given by the plane waves 7PW = E I/pX/pI' = (27r)-3/2eip'x, G Z3. Here for simplicity we assume that the Fermi ball BF := -8(0, kF) n Z3 contains exactly TV integer points, such that Tr7pw = N (put differently, N is defined via kF). Thus kF ~ (f )1/3iW3 + 0(1)^. Now we consider the interacting case. Given the regular interaction potential, it turns out that • The HF theory is good: £HF - EN is C^iV"1/3) (instead of o(iV1/3) as in the atomic case). • The HF theory is trivial: 7pw remains the unique HF minimizer. / Theorem. Assume that V > 0 and ^2k€z3{l + |/c|)l/(A:) < oo. Then PAr>^HF(7pw)+0(iV"1/3). Consequently, EUF = EN + C^iV"1/3). Proof. Let £ -^a(^3W) be an arbitrary normalized wave function. For the kinetic energy, we have n *N,22(-AX^N) = Tr(-A7« ) > Tr(-A7pw). 196 CHAPTER 9. HARTREE-FOCK THEORY Hence, it suffices to show that i 0 we can use Onsager's lemma: n E n ilv(0)N2-^V(0) l<£ lv(0)N2-^V(0). i PPW^ = x) N p£BF p£BF (2tt)3' Hence, and p7pw(x)p7pw(y)1/(x - y)dxdy = N2(2ir)-á / V = N2V(0) T3 jt3 T3 jt3 |7pw(x,y)|21/(x - y)dxdy = (27r)"6 / / IV eip-{x~v) T3xT3 1 p€BF Vix — y)dxdy (2tt)-3 í I V eip-x\2V{x)dx (2tt)-3 J E eW'X E e~iq'X E V{k)elk-Xdx p€BF q€BF fceZ3 9.5. HARTREE-FOCK ENERGY OF A HOMOGENEOUS FERMI GAS 197 (p—q+k)-x E E (27r)-3 / eV E E 5p-g+fe=o fcGZ3 P,q€BF Yv(k)\BFn(BF + k)\. fcez3 Here |-Bp n (-Bf + &)| is the number of integer points in Bp n (-Bf + We have \BF n(BF + k)\ = \BF\ - \BF\(BF + k)\>N- C\k\N2/3 Figure: Bp\(Bp + k) is the set of integer points in the grey area Thus the exchange term can be bounded from below by T3 jt3 \-/pw(x,y)\2V(x-y)dxdy > E V{k)(N - C\k\N2/3) fcez3 = N E V^ ~ CnV3 E \k\^(k) = + 0{N2'3) fcez3 fcez3 In summary, ^v, E V(xe-Xj)*N)> [ [ [p^(x)p^(y)-\^w(x-y)\2]V(x-y)dxdy+0(N2/3). i £HF(7pw) + OiN-1'3) > £HF + OiN-1'3) >EN + OiN-1'3). In particular, we have En = E + 0(N -1/31 □ 198 CHAPTER 9. HARTREE-FOCK THEORY Theorem. Assume that V > 0 and N'1/3 EfceZ3 V(k) < 1/3. Then 7pw is the unique Hartree-Fock minimizer. Proof. We use the argument of Gontier—Hainzl—Lewin (2018) where they proved that E'HT _ £HF^,pw^ ^s exponentially small for the electron gas in an infinite volume. Here since we have a bounded potential on finite box, the spectral gap of the Laplacian dominates the effect of the mean-field interaction potential. Recall that h = iV-1/3 and A = A^_1. The condition on the potential reads A||t/|Li < -h2. II IK - g Step 1. Take 0 < 7 = 72 < 1 on L2(T3) with Tr 7 = A^. Let us show that £(7) - £(7PW) > Tr((-/*2A - AG)(7 - 7PW)) -7i( I \lix^v)-jpw(x,y)\2V(x-y)dxdy. 1 Jt3 Jt3 By the definition of the Hartree-Fock functional, £:(7)-^(7pw) = Tr(-/*2A(7-7pw))-^ f f [|7(x, y)\2 -|7pw(x, y)\2]V(x - y)dxdy t3 jt3 2 rp1(x)p1(y) — p7pw(x)p7pw(x)]l/(x — y)dxdy. t3 Jt3 2 Note that p7pw is a constant since the kernel of 7pw is jpw(x — y) with 7pw(x) = (2tt)-3 £ eip-x. p€BF Hence, from the condition V > 0 and JJ3 p1 = JJ3 p7pw = N = (27r)_3p7pw(x) we find that / / [Pj(x)Pj(y) — p7pw(x)p7pw(x)]l/(x — y)dxdy > 0. Jt3 Jt3 Indeed, / / p1(x)p1(y)V(x-y)dxdy= / p7(x)p7(y) V V(k)e-lk (27r)6n0)|p7(0)|2 fcez3 = V(0)N2 = (27r)6F(0)|p7pw(A:)|2 p7pw(x)p7pw(x)V(x — y)dxdy > 0. t3 Jt3 We can rewrite the exchange term using |7(x,y)|2 - |7pw(x - y)|2 = |7(x,y) - 7pw(x " V)? + 2®(l(x,y) - 7pw(^ - y))7pw(y - x) and (7(x, y) - 7pw(x - y))7pw(y - x)V(y - x)dxdy = Tr((7 - 7pw)G) t3 Jt3 where G is an operator on L2(T3) with kernel 7pw(x — y)V(x — y). Equivalently, in Fourier space G is the multiplication operator G(k) with G(k)= £ V(k-p) p€BF because (Gf)(x)= 1^(x-y)V(x-y)f(y)dy= (27r)"3 £ e^"^(x - y)f(y)dy T3 pGBF and hence Gf(k) = (2tt)-3 / (G/)(x)e-lfc'xdx Jt3 (2tt)-6V / / e-*V^^F(x-y)/(y)ch;dy (2tt)-6V / / e-*(^)e^F(z)/(y)dxdy /r, 3 £ V(k-p)f(k). PěBf 200 CHAPTER 9. HARTREE-FOCK THEORY In particular G > 0 and hence Tr(G(7 - 7pw)) g R. Thus [|7(x, y)\2 - |7pw(x - y)\2]V(x - y)dxdy T3 jt3 = 11 |7(x,y)-7pw(x-y)|V(x-y)dxdy + 2Tr(G(7-7pw)). Jt3 Jt3 This completes the desired equality. Step 2. Next, consider the kinetic term. We find an operator A > 0 on L2(T3) such that Tr[(-/i2A - AG)(7 - 7pw)] = Tr[A(7 - 7pw)2]. The point here is that we can represent the kinetic term as a quadratic expression of 7 —7pw, similarly to the interaction energy. Since 7 and 7pw are projection we have (7 _ 7pw)2 = (7^)^(7 - 7pw)(7pw)± - 7pw(7 - 7pw)7pw, (7pw)± = 1 - 7pw. Hence, Tr[A(7 - 7pw)2] = Tr[((7pw)M(7pw)± - 7pwy47pw)(7 - 7pw)]. Since Tr(7) = Tr(7pw) = N, it suffices to find A such that (7pw)±A(7pw)± - 7pwA7pw = -h2A - AG - c0 for a constant cq g R. We will choose A to be the multiplication operator A(k) on the Fourier space, for which the latter identity becomes A(k)t(k g BCF) - A(k)t(k g BF) = h2\k\2 - \G(k) - c0, or equivalently , , [h2\k\2 - XG(k) -c0, k(EBcF A(k) = \h2\k\2 - \G(k) - c0\ = I 11 W F [-{h2\k\2 - \G{k) -c0), keBF. This choice is only possible if Co satisfies sup (h2\k\2 - \G(k)) < c0 < inf (h2\k\2 - \G(k)). 9.5. HARTREE-FOCK ENERGY OF A HOMOGENEOUS FERMI GAS 201 In fact, for every k\ G Bp, k2 G Bp we have \k2\ > \ki\ (as we assume that the Fermi ball is completely filled) and hence \k2\2 — {k^2 > 1 (since the left side is a positive integer). On the other hand, 0 < G(k) = ^V(p + k)< \\V\\ei < oo. p€BF Thus h2\k2\2 - \G{k2) - (h2^2 - XG(hfj >h2- X\\V\\ei. Hence, if A|| V||^i < h2/3, we have inf (h2\k\2 - XG(k)) - sup (h2\k\2 - XG(k)) > -h2. kcBrF V / k€Bp V / 3 We can take c0 = - inf (h2\k\2 - XG(k)) + - sup (h2\k\2 - XG(k)) 2 fce£p V / 2 k€BF ^ ' which satisfies the above condition and moreover, h2 A{k) = \h2\k\2 -\G(k)-c0\>—, VfcGZ3. 3 Step 3. Using A > h2/3 > X\\V\\ei > A||U||Loo we conclude that £:(7)_£:(7pw)>Tr[A(7_7Pw)2]_ a r r hiXjy)_^iXjy)l2Vix_y)dxdy z Jt3 Jt3 >^Tr[(7-7-)2]-^^ / / |7(x,y)-7pw(x,y)|2 ^ Tr[(7 - 7pw)2]- Thus 7pw is the unique Hartree-Fock minimizer. □ The correlation energy En — EUF will be studied in the next chapter. Chapter 10 Correlation energy Since Slater determinants are the least correlated fermionic states, the difference EN — EUF is called the correlation energy. Calculation the correlation energy is generally difficult. In this chapter, we will formulate a general framework to discuss the correlation energy, and then focus on the homogeneous gas described by the Hamiltonian N HN = 22h2(-AxJ+N-1 22 Vfa-Xj) on L2(T3W), h = iV"1/3. i=l l0, compactly supported and small enough, then EN = EUF + ECOTT + o(N~1/3). Here the correlation energy is given by ECOTT = iV-1/3 J^fc g{k) with 7tk2 — k f°° r — i / 3 \ g(k) =--V(k) + - log l + 27r/«1/(A;)(l-arctan-1(A-1)) dA, /c=( —) 2 7r J0 L J \4irJ 3 \i/3 It is convenient to use the Fock space formalism where the number of particles are not fixed. The reason is that we have to perturb the Hartree-Fock minimizer and the correlation energy will be described by the excited particles which live in a Fock space rather than in a fixed n-body space. 202 10.1. FOCK SPACE FORMALISM 10.1 Fock space formalism 203 Definition. Let L2(£l) be a one-particle Filbert space (with £1 C M.d). The corresponding fermionic Fock space is the Hilbert space oo T = F(L2(Q)) = 0 L2a(fln) = C © L2(Q) © L2a(Q2) © ... n=0 • Any vector in T has the form = (xPra)^L0 where tyn G L2(ttn) and oo ll*l& = £ll*« n=0 The vector |0) = (1,0,0,...) is called the vacuum. On Fock space, we will use the second quantization method, which goes back to Dirac (1927). A key concept is creation and annihilation operators. f Definition. For any f G L2(fl), we can define the creation operator a*(f) and the annihilation operator a(f) on the fermionic Fock space F(L2(Q)) as follows: • a*(f) : L2a(ttn) ->■ L2a(ttn+1) for all n = 0,1, 2,... ^ n+l {0*{f)^n){xi, Xn+1) = - y^(-iy~1f(x:j)^n(x1, Xj_U Xj + 1, Xn+1). Vn + 1 ^ • a(f) : L2a(ttn+1) ->■ L2a(ttn) for all n = 0,1, 2,... (a(/)^n)(xi,.. ., x„_i) = \fn \ f(x)Wn(x, xu ..., x„_i)dx. si Remarks: • y i—y a*{f) is linear, but / i—> a(f) is anti-linear. • a(/)|0) = 0 and a*(/)|0) = /. More generally, if {u{\f=1 are orthonormal functions in L2(Q), then the corresponding Slater determinant can be written as ui A ii2 A ... A un = a*(ui)...a*(un)\0). 204 CHAPTER 10. CORRELATION ENERGY For example, a*(ui)a*(u2)\0) = a*(ui)u2 = —^(ui(xi)u2(x2) — ui(x2)u2(xi)) = ui A u2. v2 The following exercise shows that we cannot put two particles in the same quantum state, which is consistent with Pauli's exclusion principle (we will come back to Pauli's exclusion principle later). Exercise. Prove that for every f G L2(tt), we have (a*(f))2 = 0 on the fermionic Fock space T{L2{yi)). The following exercise shows that a*(f) is adjoint of a(f). Exercise. Prove that for all f G L2(fl), we have It turns out that the creation and annihilation operators satisfy the following nice algebraic relations. Theorem (Canonical Anti-commutation Relations - CAR). Consider the Fock space F(L2{tt)). For all f,ge L2(n), we have {a(f),a(g)} = 0, {a*(f), a* (g)} = 0, {«(/), a*(g)} = (f, g). Here {A, B} := AB + BA. A consequence of the third relation is that a*(f)a(f)+a(f)a*(f) = Thus both a{f) and a*{f) are bounded and ||a(/)||0p < ||/||, ||a*(/)||oP < Proof. Step 1. First, let us prove that {a(f),a(g)} = 0, namely a(f)a(g) + a(g)a(f) = 0. 10.1. FOCK SPACE FORMALISM 205 It suffices to show that a{f)a(g)V. a{g)a{f)V. for any function ^n G L2{Vln) and for any n > 2. By the definition of the annihilation operator, we have (a(g)a(f)^n)(x1, ...,xn_2) = y/n(n - 1) / / g(y)f(x)^n(x,y,x1,...,xn_2)dxdy = ^n(n - 1) J J f(y)g(x)Vn(y,x,x1,...,xn-2)dxdy. The equality a(f)a(g)}$/n = —a{g)a{f)^!n follows from the anti-symmetry ^n(x, y, xi,x„_2) = -*n(y, a:, a~n_2). Step 2. Since a*(/) is the adjoint of a(/), by taking the adjoint of {a(f),a(g)} = a(f)a(g) + a(g)a(f) = 0 we find that {a*(f), a*(g)} = 0. Step 3. Finally, we prove that Similarly, w/),«*g/)} a(f)a*(g)+a*(g)a(f) = (f,g). When testing with the vacuum, we have a(f)a*(g)\0) + a*(g)a(f)\0) a(f)g-0 = {f,g). 206 CHAPTER 10. CORRELATION ENERGY Now consider any function ^n G L2a(Vtn) with any n > 1. We have (a(f)a*(g)^n)(x1,xn) = a(f){a*(g)^n)(xu xn) = \/n + 1 / f(xn+1)(a*(g)tyn)(xn+1,x1,...,xn)dxn+1 ^77+1 r_ i n+1 = Vn + 1 / - V](-l)1~1ff(a;i)^n(.Ti,x^i, xm,xn+1)drn+l J Vn + 1 tt n+l fixn+1) ^2(-l)n+l~1g(xl)'^n(x1,xn+i)dxn+i 1=1 = (/, g)^n(x1, ...,Xn) + E(-1)"+1~^(a;i) / f{Xn+l)Vn(?l,-,Xi-l,Xi+1,...,Xn+1)dxn+1. 7 = 1 ^ On the other hand, (a*(g)a(f)^n)(xu...,xn) = a*(5)(a(/)^))(xi,...,xn) 1 n = —p= y^(-l)l_1^(xl)(a(/)^„))(xi, ...,xn) v 7 = 1 I 71 r_ = —F= y^i-iy^gix^y/n / f(xn+1)tyn(xn+1,x1,...,Xi-i,xi+1,...,xn)dxn+1 i " /•_ = —j= ^i-iy^gix^y/n / /(xn+i)(-l)n_1*n(.Ti,Xi-i,xi+i, ...,xn,xn+1)±xn+1 77 „ = ^2(-l)n~lg(xl) / f(xn+1)tyn(x1,...,xi-1,xi+1,... = (x$?n)rx=0 G J7, the expectation of the number of particles is oo n=0 Let {"Uj}^! be an orthonormal basis for L2(fl). Then we can write oo M = a*(ui)a(ui). More generally we have i=i Theorem (Second quantization of one-body operators). Let h be a self-adjoint operator on the one-body Hilbert space L2{Q). Then the operator on the fermionic Fock space F(L2m oo n dT(h) := 0 ( Y hi) = 0 © h © (h © 1 + 1 ® ^) © - n=0 i=l is called the second quantization of /i. it can 6e rewritten as dT(h) = Y {um,hun)a*(um)a(un). m,n>l Here {un}n>i is an orthonormal basis for L2(tt). The representation is independent of the choice of the basis (provided that all {um, hun) are finite). Proof. Let us write an = a(un) for short. It suffices to prove that N y = E hun)a*man^N i=l m,n>l for all G Ji?®sN and for all N. Recall from a previous computation N _ r_ (a*man^N)(xu ...,xN) = E^-1)^ lumixi) / un(y)yn(x1,...,Xi-1,xi+1,...,xN,y)dy. i=i Therefore, "mi 208 CHAPTER 10. CORRELATION ENERGY 22{-l)N~l E ( E^m' hu^u^Xi^j j un{y)^N{xu xm,y)dy y)dy i=l n ^ AT 1=1 N N X\, Xi — i, X^+l, X_/Y, Xi N \N-i hN^N x1,...,xi-1,xi+1,...,xN,xi)= 7 ,\rHvN i=i Eh* = E(-1)w1(/iEi^)w)/ 1=1 AT %=i Here we have used the Parseval's identity 22(um, hun)um = hun, m the resolution of the identity operator E \Un)(Un\ = 1, n and the anti-symmetry This completes the proof. For the two-body interaction operators, we have □ Theorem (Second quantization of two-body operators). Let W be a self-adjoint operator on L2{\Tt2) such that W\2 = W2\. Then the operator on the fermionic Fock space oo ©( E w*) = o®o®w12®(w12 + w23 + w13) n=0 1<^l Here {un}n>i is an orthonormal basis for L2(tt). The representation is independent of the choice of the basis. The proof of this result is left as an exercise. Remarks: • From the method of second quantization, the typical Hamiltonian N HN = ^hl + E Wi3 i=l l un, Wup uq) . In the litterature, people also use the creation and annihilation operators a* and ax x G £1, defined by = J fi^Kdx, a(f) = J f(x)axdx, V/ G jf. These operator-valued distributions satisfy the CAR {ax,ay} = 0, {a*,a*} = 0, {ax, a*} = d0(x - y). The advantage of these notations is that we can use the second quantization without specifying an orthonormal basis for ,3^. For example, the typical Hamiltonian N HN = ^2(-AXt+V(xl))+ E Wixi-xj) i=l l 1, there exists a unitary operator R on the fermionic Fock space F{L2{VL)) such that R\0) = ui A ii2 A ... A un = a*(ui)...a*(uN)\0) and R*a*(Ul)R = Moreover, R = R* = R'1. a(ui) ifiN. Proof. The fermionic Fock space has an orthonormal basis of Slater determinants {a*{uil)...a*{uit)\0) : 1 < h < i2 < ... < ie, ^ = 0,1,2,...}. The operator R is defined by Ra*(uiA...a*(uie)\0) = a*(ujl)...a*(ujk) 10.2. PARTICLE-HOLE TRANSFORMATION 211 where 1 < j\ < j2 < ... < jk is determined from 1 < i\ < i2 < ... < it such that {ji, -,3k} = \J{is ■ > N}\J{1 < r < N : r (£ {iu ie}}. Then clearly R is a unitary operator since it maps an orthonormal basis to an orthonormal basis. Moreover, R = R* = R~x. Since the identity (a(ui) iiiN holds for all Slater determinants, it holds for any vector on Fock space by the linearity. □ Note that the transformation R is a special example of a Bogoliubov transformation. We will come back to Bogoliubov theory later in connection to the bosonic picture. Now let us focus on the homogeneous Fermi gas where N fermions are confined in the torus T3 = [0, 27r]3 (with the periodic boundary condition), described by the Hamiltonian N HN = J]V(-AXJ +N-1 V(Xl-Xj) on L2(T3W), h = iV"1/3. i=l l 0 and is compactly supported. Using the annihilation operators ap = a(up), up = (27r)-3/2eip'x we can write in the second quantization formalism (why?) HN=Y h2p2a*pap + ^ ^ V(k)a*P+ka*q-kaqaP ■ fcez3 fc,p,gez3 Also for simplicity we assume that N=\BF\, Bp = B(0, kp) n Z3. The Fermi ball Bp corresponding to the Slater determinant /\p€Bp up which is the unique Hartree-Fock minimizer. To find a correction to the Hartree-Fock theory, we apply the particle-hole transformation R\0) = f\up 212 CHAPTER 10. CORRELATION ENERGY and R*QpR — ap if p £ Bp, a; if p £ BCF. This operator really makes a hole in a Fermi ball and create a particle outside. For a warm-up, let us consider the kinetic term. We have R* i e h2p2a*pap \ R = e h2p2R*{a*pap)R = e h2p2(a*pap) + e h2p2(apa*p) P&Bp p£BF = e h2p2(a*pap) - e />2A>P + ^ e pGBp pGBF pGBF We define M0 := R* e ^Va>P fl-/i2^p2= e ^V^p) ~ e hVa*PaP- \pGZ3 / pGBF pGBJ, pGBF Note that the operator Ho does not seem positive at first sight, but it is, at least for the relevant class of wave functions. P2. Lemma. There exists a constant cq £ |N such that for every = Rty with ^ a normalized function in L2(T3N), we have M-oib = h2 \k2 — c^aLakib and inf \k2 — cq\ > —. ^—' fcGZ3 r>- fcGZ3 Proof. Note that R*MR = R* e a*Pap ) R = e a*paP + e aPa*p \pGZ3 / P€BF p€BF = e a*Pap + e ^ ~ a*pap">= N + e aiap ~ e a*p pGBJ, pGßF pGBJ, pGßF =: iV + A/"p -Mh. pap 10.2. PARTICLE-HOLE TRANSFORMATION 213 Hence, the condition = Rty with \I> G L2(T3Ar) implies that (A/"p - Mh) V = 0. Using the fact that |/c2|2 — |£>l|2 > 1 if k2 G -B^, £4 G .Bp (since the Fermi ball is completely filled) we obtain inf \k\2 — sup \k\2 > 1. Define c0 := - inf \k\2 H— sup \k\2. 2 fees- 2 fceBF Then sup \k\2 < Cq < inf \k\2, inf l/c2 — c0| > -. Moreover, using (A/"p — Mh)ip = 0 we find that M0ip = ^ h2p2a*paP4) — ^ h2p2a*paP4) = X] ^l^2 ~ co\a*Pap^- pGZ3 □ Now we turn to the interaction part. We introduce the set rnor of all momenta k = (£4, k2, k in Z3 Pi supp V satisfying k3 > 0 or (/c3 = 0 and k2 > 0) or (k2 = k3 = 0 and £4 > 0) . This set is chosen such that pnor p ^_rnor^ = ^ pnor (J (_pnor^ = j^3 r gupp ^ \ |q| _ A length but straightforward computation shows that W^r E V(k)a;+ka;_kaqap\R=^V(0)--^- £ V(p- q) + Q + X ■3 J 214 CHAPTER 10. CORRELATION ENERGY where Q = N~1 Y V{k)(b*(k)b(k) + b*(-k)b(-k) + b*(k)b*(-k) + b(-h)b(hf), fcernor X=2^ E V(k)(v*(k)V(k) + V*(-k)V(-k)) ^ Y V(k)(v*(-k)b(k)+V*(k)b(-k) + h.c.) 2N fcernor 2^ E (3 E a^ ~2 E apap ~ E atap fceZ3 h€BFn(BF+k) p€BFn(BF+k) p€BFn(BrF+k) and &*(*;) := ^ ala! pap-ki p€BcFn(BF+k) 1)*(k) := ^2 a*Pap-k - E a*Pap-k- p€BcFn(BcF+k) p€BFn(BF+k) In summary, we have Lemma. This follows from the above computations and the following expression of the Hartree-Fock energy ehf = h2 Tr(_A7Pw) + 1 f f [p7PW(x)p7Pw(y) - |7p»(x - y)\2]V(x - y)dxdy Jt3 Jt3 = h2Ep2 + ^9^-JN- E V(p-q). p€BF p,q£BF since 7pw(x -y) = (2tt)-3 ^ e^e"^ = ^ ewix-y\ p^{x) = (2ir)-3N. p€BF p€BF As we will see, the terms Ho and Q contribute to the leading order of the correlation energy (which is ~ iV-1/3) and the term X can be treated as a small error. More precisely, Ho and Q are bosonizable terms, namely they can be compared with certain quasi-bosonic operators, while X is non-bosonizable but can be removed. 10.3. ESTIMATES FOR KINETIC AND NUMBER OPERATORS 215 10.3 Estimates for kinetic and number operators In this section we derive some useful estimates for EI0 and M. We start with Lemma. Let ^> G L2(T3) be a normalized function such that HNV) < EUF + 0(N~1/3). Then the state ip = Rty satisfies Proof. In the previous chapter we have proved that N ^En^-^)>f^(o)-^(o) = fno)-^ e ?(p-EHF + 0(iV-1/3) + (^,Ho^). Consequently, if jY^) < £HF + C^iV"1/3), then , Boip) < CN~1/3. Moreover, we can write Moif) = h2 \k2 — co\a*kakilj and inf \k2 — cq\ > —. fcez3 and hence = h2 E |A:2 -c0|(^a*a^) > y E^'a>^> = f E^'^ fcez3 fcez3 fcez3 Since (, EloV') < CN-1'3 and /i = iV"1/3, we find that AA/v < CN1'3. □ 216 CHAPTER 10. CORRELATION ENERGY By a technical reason, we will focus on a well-prepared approximate ground state. Lemma. There exists a normalized function G L2(T3N) such that {^,HN^) < EN + 0(N~2/3) and that the state ip = R^> satisfies ip = l(N< CN1/3)iP, Boip) < CN~1/3. We will need the following localization technique on Fock space. The idea goes back to Lieb and Solovej (2001). The formulation below is taken from a paper of Lewin—Nam—Serfaty— Solovej (2013). / Lemma (IMS formula on Fock space). Let A be a non-negative operator on the fermionic Fock space F(L2(Q)) such that PlD(A) C D(A) and PlAPJ = 0 if \i - j\ > £, where Pl = t(Af = i). Let f,g:M.—> [0,1] be smooth functions such that f2 +g2 = 1, f(x) = 1 for x < 1/2 and f(x) = 0 for x > 1. For any M > 1 define fu := f{MjM) , gM := g(M/M) . Then ±(A- fuAfu - gMAgM) < ^[A]Ai^l{N < M + £) where Cf = \\f\\2LOO + h^h^ and [A]diag := ZZopiApi- Proof. Using the "double commutator identities" [[A, fM],fM] = fMA + Af2M - 2fMAfM, [[A, gM],gM] = g2MA + Ag2M - 2gMAgM. we have the "IMS-identity" A - fMAfM - guAgM = \ {[[A, fM], fu]] + [[A, gM], gu]]) ■ This is an analogue of the standard formula for the Laplacian (-A) - /(—A)/ - g(-Ag) = -\(\V.f\2 + |V^|2), f2 + g2 = 1 10.3. ESTIMATES FOR KINETIC AND NUMBER OPERATORS 217 which was named after Ismagilov, Morgan, Simon and Israel Michael Sigal. Next, by decomposing further oo ^ = EP* 1=0 we find that oo oo [[A, fM], /m]] = E P^A> /m], /m]]P,- = E (/m« + fM(j) ~ 2fM{i)fM{j))PiAPj i,j=0 i,j=0 oo 2 oo 2 = E Cm*) - faü)) p*Api = E (/(w - fu/Mj) PiAPj. i,j=o i<\i-j\ £. Combining with a similar formula for jm, we arrive at PiAPj. 1 oo a - fuAfu - gMAgM = ^ E [(/(W - f(j/M))2 + (g(i/M) - g(j/m)f i 0 we have the Cauchy-Schwarz inequality ±(PiAPj + h.c.) < P.AP, + PjAPj. Thus we conclude that ± (a~ ImAJm ~ guAgu^j 1 oo = ±- [(/(W - fU/M))2 + (g(i/M) - g(j/m))2] (PAP, + PjAPA i<\i-j\ 0. We can take £ = 4 as the Hamiltonian A changes particle number by at most ±4. Moreover, from the explicit formula A = M0 + Q + X + C0N~1/3 we find that N-1 E V{k)b*{k)b{k) + ± E V{k)(p\k)V{k) + V%-k)V{-k)) fcernor fc£rnor ^E^fc)(3 E ^- E a*pap) + CoN-1'3 -4diag — Ho 2N fcez3 h£BFn(BF+k) P£BrFn(BF+k) p^BcFn(BF+k) with b*(k) ■= E a>p-fc' := E apQp-fe ~ E ataP-k- p€BcFn(BF+k) p€BcFn(BF+k) p€BFn(BF+k) Let us estimate Aiiag from above. Clearly ±(3 E P = CN2'ZN pGZ: 3 10.3. ESTIMATES FOR KINETIC AND NUMBER OPERATORS 219 and ®*(k)®(k)<2 £ a*pap_ka*q_kaq + 2 £ a*pap_ka*q_kaq p,q€BcFn(BF+k) p,q€BFn(BF+k) < 4 £ a;ap_fca*_fcag = 4 £ a;(<5TO - a*_fcap_fc)ag = 4A/" + 4 £ a;a*_fcagap_fc p,qGZ3 < 4A/" + 2 £ (a*a*_fcag_fcap + a*_fca*agap_fc) p,qGZ3 = 4A/" + 2 £ (a*pNap + a;_fcA^ap_fc) pGZ3 = 4jV + 2 £ (a>P(A/" - 1) + a*p_kap_k(n - 1)) pGZ3 = AN + AN{N-l) = AN2. Thus in summary, Aiiag < H0 + AN^N2 + CiV-1/3A^ + CN~1/3. Hence, for every M ~ A^1/3 the abstract localization lemma gives ±(A- juAju - gMAgM) <^(ßo + C). which is equivalent to ±(RHNR* - fMRHNR*fM - gMRHNR*gM) < -^(H0 + C). Step 2. Now let ^gs G L2a(T3N) be a ground state for HN and denote ipgs = R^gs. Then we know that <^gs,Ho^gs) < CN~1/3, <^gs, A%s) < CN1'3. Hence, with M ~ N1^3 we have (^gs, (iZtfjviT - fMRHNR*fM - gMRHNR*gM)ipgs) > -CN~2'3. Note that (V'gs, RHNR*tPgs) = (^gs, i/Ar^gs) = Eat 220 CHAPTER 10. CORRELATION ENERGY and (ipgS,gMRHNR*gMipgS) > EN\\gMipgs\\2 = EN(1 - \\fMipgS\\2)-Thus we conclude that <^gs, fMRHNR*fM^gs) < EN\\fM^gs\\2 + CN~2I\ Note that we can choose M = CN1'3 > 4(^gs,A/V>gs), which ensures that HWgsll2 = {^gs,g2M^gs) < {^gs, ^Ags) < \i namely Finally, we define \\fM^gs\\2 = 1 - ll^ilW'gsll2 > \- I filings .j, D , ^ = II f /IP ^ = ^v^- Then we have HN^) = RHNR*i)) p- E a>p) fcGZ3 h£BFn(BF+k) p€BcFn(BF+k) p€BcFn(BF+k) 10.4. REMOVING THE NON-BOSONIZABLE TERM 221 where b*(k) ■= £ a*pa*p-k, ®*(k) ■= £ a*paP-k - £ a>p-fc. p€BcFn(BF+k) p€BcFn(BF+k) p€BFn(BF+k) Lemma. Let = R^ with V G L2a(T3) and > ip = t(M< CN1/3)ý, I0ý) < CN~1/3. Then we have o(N~ -1/3) Proof. Step 1. The third sum in X is easy since ±N~1(3 £ a*hah-2 £ a*pap - £ a>p) < 3JV"W h€BFn(BF+k) p€BcFn(BF+k) p€BcFn(BF+k) and N'1^,^) < 0(N~2/3). Step 2. Now let us prove that for every O^fce Z3, ~ 0(1), N~1{^,1)*(k)1)(k)ip) < o(N~1/3) « {ip,V*(k)V(k)ip) < o(N2/3). Note that we have proved 1)*(k)1)(k) < AM2. However, from that bound and the a-priori estimate on AT we only have (ip,D*(k)D(k)ip) < 4,.A/V) < GiV2/3. To obtain the refinement o(iV2/3) we need a better approach. By the Cauchy-Schwarz inequality (A + B)*(A + B) < 2{A*A + B*B) we have {iP,Q*(k)®(h)iP) <2 £ {^,a;ap_ka*q_kaq^) p,q€Z3 = 2 £ 0, Bp>0, 2y/ApBp > 1. Combining with the condition ip = l(Af < CiV1/3)^ we have | ^2{^,a;a*q_kap.kaq^) \ < CN1'3 ^ {Ap\\apiP\\2 + Bp\\ap.k^\\2) Z3 pGZ3 To estimate further the right side, we split the sum into two parts: Xi = {p £ Z3 : max{|p2 -c0\,\(p- k)2 - c0\} > £}, X2 = {p£ Z3 : maxflp2 - c0|,\(p - k)2 - c0\} < ('.} with a large parameter I (eventually we can take £ = iV2/15). Here the constant c0 is taken from the representation of the kinetic operator HoV-" = \k2 — c0\akakip and inf \k2 — c0| > -. fcGZ3 feeZ Part 1: For p £ X\ we choose A \P2 -c0\ _ \(p — k)2 — c0| p £1/2 ' UP £1/2 Then 2 APBP > 1 because min{|p2-c0|,|(p-£;)2-co|} > 1/2 10.4. REMOVING THE NON-BOSONIZABLE TERM 223 and max{\p2 - cqI |(p - k)2 - c0|} > L Thus using the kinetic bound WM) = h2Y \p2 - coKKsV) < cn-1'3 p£Z3 we have 1 CN1'3 E (a>imh2 + Bp\\ap_k^\\2) < E \p2 ~ coW'a;^) < £1/2 Part 2: For p 6 I2 we simply choose Ap = Bp = 1. Using the fermionic property ||ap||0p < 1 we obtain E {Ap\\apiP\\2 + Bp\\ap_k^\\2) <2\X2\ We can show that (see below) X2\0. Thus in conclusion, E a*pa*q-kaP-kaq^) p,q€Z3 / AJl/3 ^CiVl/3 (7172-+^ By optimizing over £ (e.g. taking £ ~ TV2/15) we find that E <^aX-feaP-feM'> 0. 224 CHAPTER 10. CORRELATION ENERGY Proof. Note that for p G X2 by the triangle inequality \2p ■ k\ = \p2 - (p - k)2 + k2\ < \p2 - c0| + |co - (p - k)2\ + k2 < 2£ + k2 < A£. Hence, X2 C {p G Z3 : \p2 - c0| <£,\p-k\< 2£}. The desired inequality |A2| < C£2N£ follows from the fact that for every given r, s G Z, lr — co| -\ £■, \s\ < 2£, we have {p G z3 : p2 = r, jj • k = s} < CN£, Ve > 0. We will need a fundamental fact from number theory. Theorem (Integer points on ellipses). Let do G N. Then when M —> oo, the equation m2 + don2 = M has at most 0(M£) solutions (m,n) G z2, for every e > 0. This result was proved by Cilleruelo and Cordoba [Lattice points on ellipses, Duke Math. J., 1994]. Let us accept it and conclude the counting argument. Easy case: Assume k = (£4, 0,0) with k\ ^ 0. Then condition p ■ k = s determines pi uniquely and for every given p\ the equation 2,2 2 P2 + P3 = r ~ Pi as at most 0(N£) solutions {p2lp:i) G z2 for every e > 0. Here we used \r — p\\ < CN2^3. General case: Now we turn to the general case. We can assume that k = (ki,k2, k^) with k\ 7^ 0 and (k2, k^) ^ (0, 0). Then we use the fact that the following vectors are orthogonal in R3: k = (A4, k2, k^), k± = (0, — &3, k2), k'± = {—k\ — k2, k±k2, k±k^). 10.4. REMOVING THE NON-BOSONIZABLE TERM 225 k 2 k± 2 k>± P'W\ + \k±\ + Using the orthogonality of (k, k±, k'±) we can write Combining with \k'±\ = |A;||A;j_|, we obtain W±\2\p\2 = \k±\2\p ■ k\2 + \k\2\p ■ k±\2 + \p-k'±\ Every p £ Z3 is determined uniquely by (ni, n2, n%) £ Z3 with ri\ = p ■ k, 77-2 = p ■ k±, 77-3 = p ■ k'±. In particular, the constraints can be rewritten as k ■ p = s, p2 = r I / 12 2 2 it/ |2 17 |2 2 ni = s, \k\ n2 + n3 = \k±\ r — \k±\ s . For every given r,s £ 0{kp), the second equation has at most 0(Ne) solutions (tt-2, n^) £ Z2. This completes the proof of \X2\ < CC2N£. Thus Step 2 is finished. □ On the counting problem on ellipses. Here is a proof of the counting problem on circles. Proof. We need to prove that for M —> oo the equation x2 + y2 = M has at most 0(Me) solutions (x, y) £ Z2. In this case, the number of solutions, denoted by r2(M), can be computed explicitly. For every MeNwe can write uniquely M = 2a -m- Y[pPp v where the last product is taken over prime numbers p = 3 mod 4, and m is the product of (powers of) primes = 1 mod 4. Then r2(M) = { if m is not a square if m is a square. 226 CHAPTER 10. CORRELATION ENERGY Consequently, we get the upper bound r2(M) < 4 J](1 + /3P Jv) v over prime numbers p = 3 mod 4 which are factors of M. We can divide the product into two parts. Part 1. If p < K for a large constant, then we simply bound (3p by logM and get n(i+&)<(ciogM) p K, then using 1 + /3P < eh < K^/losK < p^/l°sK we find that In summary, we obtain n (!+pP) < n p^p/iogK ^ m1"0^. p>K p>K r2(M) < (ClogM)KM1/losK for any K large. This implies that r2(M) < 0(Me) for any e > 0. □ Step 3. For the second term of X, we use the Cauchy-Schwarz inequality ± (1)*(k)b(-k) + h.c.) < e~11)*{k)1){k) + eb*(k)b(k). We need to prove that the expectation against is o(iV2/3). We have proved that ,2r(fc)2)(A;)V') < o(^2/3)-Therefore, it suffices to show that {ip,b*(k)b(k)iP) < CN2/3 and optimize over e > 0. The latter bound follows from the kinetic bound {ip,Moip) < CN~1/3 10.4. REMOVING THE NON-BOSONIZABLE TERM and the following lemma. 227 Lemma. For every 0 ^ k G Z3, \k\ ~ 0(1) we have This inequality was first proved by Hainzl—Porta—Rexze (2018). Below is a simplified proof. Recall that b*(k)= Y aX-*> ^ = Y.h2\p2-c^a>^- p€BcFn(BF+k) pGZ3 Proof of the lemma. As in [HPR], By the triangle and Cauchy-Schwarz inequalities we can write WKkW < ( E IK-^VAII Kp€BFn(BF+k) <( E \P2 - (P - k)2\\\aP-kaM2\ \ E 1 21 |c0 - p2\\\ap_kap^\\2 co -/HIvAII2 ypGB=n(BF+fc) ) \p€BFn{BF+k) U Vl ' The first term can be bounded by the kinetic energy E \P2 - {P - k)2\\\ap-kapip\\2 p€BcFn(BF+k) = E \{p - k)2 - co\\\ap_kaPi)\\2 + ^ p€BcFn(BF+k) P€BcFn(BF+k) < Y \(P ~ k)2 ~ c0\\\ap_k^\\2 + Y p£BFn{BF+k) P€BcFn{BF+k) N1/3 for all p G Bp n (Bp + k), then we are done. However, \p ■ k\ may be small (e.g. 0(1)) and we have to count these terms carefully. We write \p2 — (p—k)2\ ^—' Isl + 1 p€BcFn(BF+k) u K1 ' 1 Isl^CW1/3 1 1 where Bs = {p (E Bp n (Bp + k) : p ■ k = s}. We count |SS| using a similar idea of counting \Xz\ in Step 2. First try. We can write Bs = \jBs,r, Bs,r = {peBpn(BF + k):p-k = s,p2 = r}. r Here the condition p G Bp n (Bp + /c) means p2 >k2F> (p- k)2. When p ■ k = s, p2 = r, it is equivalent to k2F + 2p-k-k2 > p2 >k2F k2F + 2s-k2 >r> k\. Thus for any given s, we have at most C(\s\ + 1) choices of r. Moreover, for given (s,r), we have \BS:T\ < 0(N£) for every e > 0, by the same argument as in counting \X2\. Thus we conclude that \Ba\ < C£N£(\s\ + 1), Ve>0. From this bound we can obtain E 12 / M2|< E ^4<^1/3+£, V£>0 ^ P 2-(p-k)2\ ^ \s +1 p€BFn(BF+k) u K1 ' 1 Isl^CW1/3 1 1 which is close to the desired bound O^1/3), but not enough. Second try. Let us proceed differently, using the following 10.4. REMOVING THE NON-BOSONIZABLE TERM 229 Theorem (Integer points in ellipses). Let d0 G N. For every R > 0 consider the ellipse E(R) = {(x,y) G R2 : x2 + d0y2 < R2}. Then the number of integer points in E(R) is S(R) = \E(R)\+0(R2/3)R^00. When d0 = 1, this is the bound from the Gauss circle problem that we discussed before. The result for ellipses is slightly more difficult, but could be obtained by a similar argument (in general, the bound 0{R2^3) holds for any convex set, and it is optimal). Now let us show that \Bs\ p2 > k2F ^=> k2F + 2s - k2 - p\ > p\ + p\ > k\ - p\. The number of integer solutions (p2,pz) G Z2 is equal to the integer points in the annulus B{0,R2)\B{0, i?i) with Ri = ^k2F-pl R2 = ^k2F + 2s-k2-p2. Note that Ri p2 > k2F \k'±\2(k2F + 2s - k2) > \k±\2s2 + |£;|2772 + 772 > |^|2A;2 Thus (772,773) £ E(R2)\E(R1) where E(R) is the ellipse e(i?) = {(x, y) £ M2 : |£;| V + y2 < i?2} and Ri = y/\k'±\2kF ~ l^l2s2' ^2 = y Kl2(^ + 2s - P) - \k±\2s2. Note that Rx < R2 < CN1'3 and - i?2 < C{\s\ + 1). Hence, by the Gauss counting-problem on ellipses, the number of integer points (772,773) £ Z2 in E(R2)\E(Ri) is |e(i?2)| - \E(R1)\ + 0(f?2/3) = tt|l - tt|j + 0(a^) < c(|s| + ^ Thus in conclusion, we have proved for all \k\ ~ 0(1), \BS\ < Omin{iV£(|s| + 1), \s\ + N2/9}. Therefore, \p2 - (p - k)2\ ~ 2-^ \s\ + 1 p€BFn(BF+k) u K1 ' 1 |s|0, compactly supported and small enough, then EN = E + Ecorr + o(N -i/3\ Here the correlation energy is given by ECOTT = N x/3 ^fc g(k) with g{k) = -—V{k) + - I log l + 27TKl/(A;)(l-arctan-1(A-1)) 2 7T J0 The discussion in this section is only heuristic. We consider the bosonizable term = YJh2\p2-k2F\a*pap+^- E F(A;)(6*(A;)6(A;)+6*(-A;)6(-A;)+6*(A;)6*(-A;)+6(-A;)6(A;) Recall that fcGrn b*(k) = E p€BcFn(BF+k) apap-k- For any given k and p 6 5^ n (Bp + /c), we think of the operator b*p{k) := a*pa*p_k as a bosonic creation operator. The reason is that it satisfies and [h*P{k)> = a*pa*p-ka*qa*q-k ~ a*qa*q_ka*pa*p_k = 0, [bp(k), bq{k)] = 0 [bp(k),b*q(k)} = ap_kapa*qa*q_k - a*qa*q_kap_kap = ap_k(5pq - a*ap)a*_k - a*(5pq - ap_ka*_k)ap = bpq{ap-ka*q_k - a*ap) - ap_ka*apa*_k + a*ap_ka*_kap = 5pq(ap_ka*_k - a*ap) 232 CHAPTER 10. CORRELATION ENERGY = Spqi1 ~ a*P-kaP-k ~ a*pap) = o~Pq - 6pq(a*_kap-k + a*ap). These relations look similar to approximate CCR for bosonic operator. The error term 5pq{a*p_kap-k — apap) m [°p(k), b*(k)] is not identically equal to 0, but it is small in average since E aiaP^) < < CNl/3 < N2/3- p€BcFn(BF+k) Similarly, we can show that if k ^ £, then [b*p(k), b*q(£)] = 0, [bp(k), bq{£)] = 0, [bp(k), b*q(£)] « 0. This means that the different momenta k correspond to different Fock spaces. This is consistent with the random phase approximation developed by Bohm—Pines (1960s). Thus the interaction term Q = ^ 22 V(k)(b*(k)b(k)+b*(-k)b(-k)+ b*(k)b*(-k)+ b(-k)b(k)^ fcernor = jf E 9^ 22 ib*P(kMk) + b*p(-k)bq(-k) + b*p(k)b*q(-k) + bp(-k)bq(k)) fcernor p,q€BFn{BF+k) looks like a quadratic Hamiltonian in a bosonic Fock space. It is somewhat less obvious that the kinetic operator is also quadratic in terms of b*(k) and bp(k). Heuristically, Mo = e h2\p2 - k2F\a>p ^ 22 22 - (p - k)2K(kMk) =■ fio- pGZ3 k£Tn°r p£BcFn{BF+k) Indeed, it does not hold in general, but it holds for a class of quantum state close to the ground state. Our key observation is that p0, K(k)] = 22 22 h2\p2- *fI[«>p, «-k\ pGZ3 q£BFn{BF+k) = 22 22 ^P2 - kl\ {Kap> a*qK-k + a*qKaP' a*q-k\) pGZ3 q€BcFn(BF+k) = 22 22 h2\p2 -k2F\ (Spqa*qa*q_k + 5p,q-ka*qa*q_k) pGZ3 q€BcFn(BF+k) 10.5. DIAGONALIZATION OF BOSONIZABLE TERM 233 = E h2(\q2-k2F\ + \(q-k)2-k2F\)a*qa*q_k q€BcFn(BF+k) = h2(q2-(q-k)2)b*q(k) while Po, K(k)] = E E E h2(p2-(p- £)2)[b*p(£)bp(£), b*q(k)] Pernor peBc n(BF+f) q€BFn(BF+k) ^ E E E " (P" 02) 0. Thanks to the linearization of the kinetic operator, the difference EIq — EIq is mostly invariant 234 CHAPTER 10. CORRELATION ENERGY under the Bogoliubov transformation. Indeed, e-B(Mo-Mo)eB-(Mo-Mo) = J - {e-tB(U0 - H0)etB} dt = [ e-tB[m0-m0,B]etBdt Jo This term is small because [Ho—Ho, B] 0, which follows from the fact that [Ho—Ho, b*(k)] ~ 0. Thus in summary we have e-BR*HNReB « EUF + ECOTT + H0 - H0 + ^ ^ Ap(k)b*p(k)bp(k) fcernor P£BcFn(BF+k) For an upper bound, we can apply the above operator inequality for the vacuum and find that EN < {0\e-BR*HNReB\0) « EUF + ECOTT. The lower bound is more difficult as we have to estimate Ho-H0+ Y E Ap(k)b*p(k)bp(k) fcernor P£BcFn(BF+k) from below. At this point, we need the smallness condition on the interaction potential V. Note that Y E Ap(k)b;(k)bp(k)-u0 = Y E (A.W-^V-(p-fe)2))fe;(*(fe)- fcernor peBc n(Bp+fc) fcernor P€BFn{BF + k) When F is small, then Ap(ifc) - h2(p2 -(p- k)2) < eh2(p2 -(p- k)2). Hence, we can conclude using the operator inequality Y h2(p2 -{p- k)2)b*p(k)bp(k) < CH0 p€BcFn(BF+k) which can be proved similarly to the kinetic inequality in the previous section. Chapter 11 Stability of matter We consider a sample of ordinary matter composed of N quantum electrons and M classical nuclei located at {Rk}^=1 C R3. The system is described by the Hamiltonian N N M % = Ba)-EE^+ E ]x-z-x-\+ E 1=1 1=1 k=l 1 1 Kkj-CZ(M + N), VM,N. Once it is done, the existence of thermodynamic limit follows easily from a general argument based on the sub-additivity E(Mi + M2, JVi + M2) < E(MU Ni) + E(M2, N2). The stability of matter was first proved by Dyson and Lenard (1967). In 1975, Lieb and Thirring gave a very short proof, using their kinetic inequality *,^(-AxA*\>K3J^p*H5 x)"/3dx and Teller's no-binding theorem in the Thomas-Fermi theory. Another route to the stability due to Solovej is to use the Lieb-Thirring inequality together with Baxter's electrostatic inequality. These approaches will be discussed in this chapter. 11.1 Teller's no-binding theorem Take M nuclei located at {Rk}^f=1 C R3 and with the nuclear charges {Zk}%L1, Zk > 0. Denote the nuclear potential n*) = -£ M „ \x — R fc fc=i Let us consider the Thomas-Fermi functional, with a constant cTF > 0, £vF(p) = [ (cTFp(xf3 + V(x)p(x))dx + i P{x)p{y)-dxdy ■ ^ Z^ x — y\ z-~/ \Rg — Rk\ y' !<£ E^on^)- fc=l 238 chapter 11. stability of matter By induction in m, we only need to show that ETF({Rk}^ (Mf=i) > £TF({i4}f^\ {zk}£?) + eIUZm). Denote M M-l „ „ V(x) = -^r^- = VA(x) + VB(x), VAx) = -YvArr V* = -T^ir-r z—' X — Kk\ z—' X — fife X — fc=l 1 1 fc=l 1 1 1 1 It suffices to show that for every 0 < p G L1(1R3) n L5/3(IR3) (we can also assume p > 0 everywhere), we can find two functions g,h > 0, g + h = p such that ^(P)>«+©)- For the kinetic energy, the condition g + h = p immediately implies the pointwise inequality (g + h)5/3 > g5/3 + h5/3, and hence p5/3 > / ^5/3 + / ^5/3_ Thus it remains to compare Coulomb potentials. We need to find g + h = p such that Iff p(x)p(y), sr^ z?zk [ p(x)V(x)dx + - [ [ 4^#dxdy+ y Jr3 2 JR3 Jk3 \x — y\ ^TTZl ,Rp — Rk l<£-[3(x)vA(^ + U [ gP^^y+ £ z,z" ■jr3 1 jr3 jr3 \X ~ V\ ts^usj 1 r r h(x)h(i 2 Jr3 Jr3 \x — y \Rp — Rk l<£ 0. Jr3 Jr3 Jr3 Jr3 \x ~ V\ \ 0. To construct g and h, we need the following special version of Baxter's electrostatic inequality. Exercise. Let 0 < p G L1(R3) n L5/3(IR3). Then there exists 0 0 everywhere.) Now we can conclude the proof of no-binding theorem. We choose g as in the above exercise and take h = p — g > 0. Then 2D(g — mA, h — mB) = / (|x| 1*g — \x\ 1*mA){h — mB) Jm? = +/ =0— / (\x\~x * g — \x\~x * rnA)rnB >Q. J9<9 J9=9 J9=9 □ 11.2 First proof of the stability of matter Now come back to the Hamiltonian on L2(lR3Ar): N , 7 7 M „ /w^-^+to E ^+ E ^ v^-E^ 1=1 KkjXW 1 n !<£ -C{M + N). First proof of the stability of matter. Take a normalized wave function ^ G L2a(R3N). Then ZeZk i=l 17 M Ki 0. Moreover, by the Lieb-Oxford inequality ii/ / f A"-cf p. z Jm? Jm? \x ~ y\ z Jm? Jm? Hm,n^) > — / + / + - / / —;-;—dxdy ^ Jr3 JR3 z Jr3 Jr3 \x — y\ + y , ^Zfc ,-CN —' \R( — Rh\ l<£v) - CN. By Teller's no-binding theorem, M M k=l k=l Thus we conclude that N'. //•:,. \ M';- > CMZ7 3 - C'A. Optimizing over ^ G L2a(R3N), the lower bound for E(M, N) follows. □ 11.3. BAXTER'S ELECTROSTATIC INEQ UALITY 241 Remark: In the original proof of Lieb and Thirring (1975), they did not use the Lieb-Oxford inequality They bound the indirect energy by Teller's no-binding theorem as follows. From g / [ f p(x) , + 1 / / toWMv)dxdy+ y , z-cyzi'\ taking Zk = 1, M = N, Rk = xk, p = py with ^ G L2a(Rm) we have z it3 l=1 z jb3 7r3 \x - y\ i G L2(lR3Ar), we obtain which is equivalent to l 0. Then El v ^ v ^ Z v ^ Z2 ^ v ^ *2Z -\-1 \x,-xA ^^\xi-Rk\ ^ \Rp-Rk\~ ^ Ttixi) l<*>• l<£ o. ^ \x-Rk\s k ' ' 244 CHAPTER 11. STABILITY OF MATTER Here nk ■ (x — Rk) > 0 on x G dTk since Tk is convex. □ Step 2. We have the following basic electrostatic inequality. Lemma. Let n be a measure with D(/i,/i) < oo. Then D((jl, (j,)- [ $(xMdx) + V ,„Z\ , > 0. i ^777^*, \B-e — rCk\ !<£ -D(u, v) since D(/i — u, ji — v) > 0. It remains to calculate D(u, v). Using the equation $ = |x| 1 * v and the fact M n, M „ M fc=i 1 1 v 7 fc=i 1 1 fc=i we can write M v , I 2 \f $(x)dv(x) < ^-J2 [ [ S(y - R^-^—rdy^x) z 7r3 ^ ^ ji3 «3 \x — y\ if; / S(y-Rk)(f -j—du(x))dy = ^f2f <% " ^ |E*<*)= e 2 ^—' ' \Rt — Rh\ k=l l<£ 2)(xj)/2 for every x G supp/Zj. We apply the above lemma £>(/i, /i) - /" $(x)p(dx) + ^ ^ > 0 ^ i 'S)(xi)/2 for every x G supp^j, by Newton's theorem we have f N f M N f Z N f Z / $(x)dfj,(x) = E / $(a;)d^(^) = EE / I p i^^) - E / ^77Td^(x) z=l fc=l z=l z=l M N z " r z EEt^^-E/^^ 1=1 k=l 1=1 M N Z A 2Z fc=l 1=1 'X; Thus in summary Z2 0 < /i) - J $(x)(j,(dx) + ^ l<^-Rk\ ^ Ste) l -C{M + N). Second proof of the stability of matter. Step 1. We consider the simple case where Zk = Z for all k = 1, 2,N. We use Baxter's electrostatic inequality 1 l •ill 1 ' i=l fc=l Z2 •*■ i —' Rf — Rk 1 !<£ AT ^ Si, i=i ^ where £)(x) = mini E (-^. 1=1 v 2Z + 1 Hence, by Pauli's exclusion principle and the Lieb-Thirring inequality, for every /i > 0 we have Hm.n > E (-^. 1=1 v 2Z + 1 Sfc) > Tr A 2Z + 1 -A - , + /i S(x) 2Z + 1 Dfx) /iN [i 5/2 dx — jiN. Since £)(x) = mini —C(M + N). Actually by optimizing over H > 0 we find that E(M, N) > -C{2Z + 1)2M2/3N1/3. Step 2. Now we come to the general case when Zk < Z for all k = 1, 2,M. The proof in this case follows from Step 1 and the following monotonicity in nuclear charges. Lemma. Denote E(M, N, {Zk}) be the ground state energy of Hm,n with given nuclear charges {Zk}. If Zk < Zk for all k = 1,2,M, then E(M,N,{Zk})>E(M,N,{Zk}). This observation is due to Daubechies and Lieb (1983). Proof of the lemma. Note that for every I 6 {1, 2,M}, the mapping Z^ —>■ HM^N is linear. Therefore, the mapping Ze^E(M, N,{Zk}) is concave (the concavity holds separately for each Ze, not jointly for {Zk}). Under the condition 0 < Z? < Z? we can write Ze = t-0 + (l-t)-Ze, for some t G [0,1]. Hence, the concavity implies that E(M, N, {Zk}) > IFaM. N, {Zk})]Zi=0 + (1 - t)E(M, N, {Zk}\Zl=~z, 248 CHAPTER 11. STABILITY OF MATTER On the other hand, setting Zg = 0 is equivalent to putting Rg at infinity, and hence E(M, N, {Zk}){Zi=0 > E(M, N, {X,})%_ x. Thus lAM.\.{Zh})> IaM.X.y. By induction, we find that E(M,N,{Zk}) >E(M,N,{Zk}). This completes the proof of the lemma. □ From the condition Zk < Z for all k = 1, 2,M and the lemma, we find that E(M, N, {Zk}) > E(M, N, {/.. /..Z}). By Step 1, we have E(M, N, {/.. /..Z}) > -CZ(M + N). This implies the same lower bound for E(M, N, {Zk}). Thus E(M, N) > -C(M + N). □ 11.5 Existence of thermodynamic limit Consider the Hamiltonian on L2(lR3Ar) N N M „ „ „ i=l i=l k=l 11 K| l R is bounded and convex (consequently it is continuous). Proof. Step 1. We have the following sub-additivity E(Mi + M2, JVi + N2) < E(MU Ni) + E(M2, N2). This is an easy consequence of the variational principle. More precisely, given two wave functions tyNl G Z^(]R3Ari) and ^N2 G Z^(]R3iV2) we can construct a trial wave function \I>^ N2 in L2,(R3(Nl+N2^) by antisymmetrizing the product ^Nl(xx, ...,xNl)^/N2(xNl+1 + y,...,xNl+N2 +y). Then E(M1 + M2, JVx + N2) < lim E(M, N) is decreasing. From the stability of matter, we obtain E(M + N) ~ M + N ~ Hence, for every r\ G (0,1), we can find a sequence (Mj, Nj) such that w ,r ^ , E(Mj,Nj, Mj.Nj —> oo,----> 77, hm —-—-—— exists. 31 3 ' Mj + Nj ' j->°o Mj + Nj 250 CHAPTER 11. STABILITY OF MATTER N'- It remains to show that for any (M-, iV') with iV- —> oo, m'-In' ~* V we a^so have lim -—^-v- = lim Indeed, by passing to a subsequence of (M-, iV') if necessary, we can assume that M'3/M3 —>■ oo and N'j/Nj —> oo. Define frMJi riV.'il G N := min M3 5 where [t] is the integer part of t (i.e. [£] , N>) < E(LjMj, L3N3) < L3E(M3, N3) L3{M3 + Nj) E(M3, N3/ M'3+N'3 ~ M'3 + N'3 ~ M'3 + N'3 M'^+N'j M3 + Nj Thus E(M3, Nj) E(M3,N3) lim sup ——— < lim m; + n; -,■->«, Mj + Nj Similarly, by passing to a subsequence of (Mj, Nj) if necessary, we can assume that M3/M'3 oo and N3/N'3 —> oo. The same argument as above E(M'3, Nj) E(M3,N3) In summary, Hence, the limit lim inf-—^-— > lim j->oo M'3 + N'3 ~ j^oo M3 + N3 E^N'i)-v^ E(M3,N3) lim , rl „T, = lim ,, . j->°0 Mj + iVj j->oc Mj+Nj e(ri) := lim —-—-—- V 1 M,N^oo M + N N/(M+N)^r1 exists. Step 3. Since 0 > E(M + N) > -C(M + N), we have 0 > e(rj) > -C 11.5. EXISTENCE OF THERMODYNAMIC LIMIT 251 for every 77 G (0,1). Thus e{rj) is bounded. The convexity of 77 —> e{j]) follows from the subadditive of E(M, N). Indeed, let 77,77' G (0,1) and let N,M = M(N), M' = M'(N) ->■ 00 such that N N Vi *„ , Ar V ■ M + N 11 M' + N Then N(M + N) + N(M' + N) 1 f N N \ 77 + 77' 2(M + N)(M' + N) 2 \M' + N M + N/ Note that 2(M + N)(M' + N) - N(M + N) - N(M' + N) = NM + NM' + 2MM'. Hence, £ (Vm + JVM' + 2MM', N(N + M) + JV(JV + M')) (rj + rj' 2(M + iV)(M' + JV) ^ 6 V 2 On the other hand, by the sub-additivity of the ground state energy, we have E [NM + NM' + 2MM', N(N + M) + JV(JV + M') < E(NM, N2) + E(NM', N2) + E(MM', NM) + E(MM', NM') < NE(M, N) + NE(M', N) + ME(M', N) + M'E(M, N) = (m' + N)E(M, N) + (M + N)E(M', ao-Dividing both sides by 2(M + JV)(M' + iV) we obtain £ (Vm + a^m' + 2MM', N(N + m) + JV(./V + my 2(M + N)(M' + iV) < 1 fE(M,N) t E(M',N) 2 \ M + N M' + N Taking the limit we conclude that 77 + 77 < I (e(v) + e(r?')) • 2 y ~ 2 Since 6(77) is uniformly bounded in (0,1), the latter bound implies the convexity, namely e ((1 - t)rj + trf) < (1 - t)e(rj) + W G (0,1). □ 252 CHAPTER 11. STABILITY OF MATTER Exercise. Let f : (0,1) —> R be a bounded function such that /(^<^!, va.»e(o,i). Prove that f is convex in (0,1). 11.6 Grand-canonical stability There is also the stability in the grand-canonical setting, where the ground state energy is computed without the particle number constraint, but with a volume constraint. For simplicity, let us consider a system of N particles of charge —1 and M particles of charge +1 in an open bounded set C R3. The system is described by the Hamiltonian N M N M Vk> ^ \xx\ jL^ \y y\ JL^ JL^ \X y\ i=\ k=l 1 L2a(flM). The grand canonical ground state energy is E(Cl) = inf inf (V,HMnV)- M,N -C\Q\ with a finite constant C > 0 independent oftt. This result holds under a more general assumption, where the masses and the charges of the particles can be different. Proof. By the canonical stability we have i=l l-M5/3 with a constant K > 0. Thus in summary, Thus > mf (T^(M5/3 + iV5/3) - C(M + iV)) > □ Note that the energy E(tt) satisfies the following properties: • Translation-invariant E(Cl + z) = E(Cl) for all z G R3. • Sub-additivity U Q2) < + £(^2) if ^1 n Q2 = 0. • Stability E(Cl) > -C|fi|. All that implies f Theorem (Existence of thermodynamic limit). The limit lim B(n) si=[-l,l]3 |ri exists and it is finite. The proof of this theorem is left as an exercise. Actually the existence of the thermodynamic limit holds for a much bigger class of domains fl. For example, tilling domains are allowed. 254 CHAPTER 11. STABILITY OF MATTER 11.7 Instability for bosons In the proofs of the stability of matter, the fermionic property is crucial. Indeed, the stability fails if Pauli's exclusion principle is turned off. Let us consider the Hamiltonian n n m „ „ „ i=l i=l k=l ' 1 K| l 0 independent of N. The lower bound was proved by Dyson and Lenard (1967). The upper bound was proved by Lieb (1979). Proof. Lower bound. By Baxter's electrostatic inequality n m 2 n „ \xi-xA \x% - Rk\ ^ \Ri-Rk\~ 5)(xi) l -CN2'3 on L2 (R3). 11.7. INSTABILITY FOR BOSONS 255 By Sobolev's inequality (c.f. CLR bound), there exists eo > 0 such that for every ji > 0 satisfying 3 we have Sfx) -A 3/2 Dfx) dx < Eq. ■fi> 0. By the definition of S(x), we can bound n -i 3/2 n Dfx) dx < V I k=l = N |x — Ru 3/2 dx Ti ~ V x 3/2 dx = CNß -3/2 Thus the condition CNji 3I2 < eq is satisfies when ji ~ N2^3, as desired. Upper bound. We take the trial function ^(xi,xtv) = u®N(xi,xtv) = u{xi)u{x2)...u{xn). with a normalized function u G L fR ). Then EB{M, N) < (u®N, HMiNu®lv) = N I \Vu N(N - 1) N \u(x)Y n E / , r~~f ./i»3 \x — Ri i=i rdx \u{x)\2\u{y)\' \x-y\ 1 dxdy —' \Rp — Ru Note that this upper bound holds for any choice of the nuclear positions {Rk}k=i- Hence, we can average over {Rk}k=i- First try. Integrating the above variational inequality against \u{R1)\2...\u{RN)\2dR1...dRN, Rk G M3 we obtain EB(M,N) i-r ^ Jm3 \x - Rk\ \Re-Rk\ against Vlu^i)!2)...^^^)!2)^!...^, Rk g Qfc we obtain f , ,9 N(N-l) f f \u(x)\2\u(y)\2 , , EB(M,N) 0 we obtain the desired upper bound EB(M,N) < -C_1iV5/3. □ In the above we have ignore the kinetic energy of the nuclei. The situation changes a bit when we consider nuclei with finite masses, however the instability remains. In the following let us consider the Hamiltonian N N N N ^ = £(-A-) + E(-A-)-EE^^ E + E rJ—r- \x% — Vk\ \Xi—xA z—' \yp — Vk\ 1=1 k=l 1=1 k=l 1 y 1 Kkj £ = iV1/5. Thus Evrf'sfa f \Vu0\2-I0 f \ jr3 Jr3 M5/2 Optimizing over u$ gives us the desired upper bound. Step 2. It remains to find a trial state in the 2iV-particle sector. We can repeat the choice of the trial state # = WU|0) but now we use the Weyl unitary transformation W = eVM(a*(u)-a(u))^ 0 < u E (R3), ||u||L2 = l with M = 2N — CN3/5, so that = M + Tr7 < 2N - N3/5. Recall that with our choice of 7, the number of excited particles is Tr 7 = 0(N3^5). Moreover, we have Ai2^) - {V,M^)2 = (v, (M - {V,M^2^ < CN. Similarly to Step 1, we have E = \ ^>($Hn > < nV5 inf / (2\Vu(x)\2-I0\u(x)f2)dx + o(N7/5). \ n=Q / ll«llx,2(R3)=i 7r3 11.7. INSTABILITY FOR BOSONS 261 At the moment, \I> is still a state on Fock space. To go to fixed particle sectors, we denote 2 /n>3ra\ £ll^||^„||2EB(n). \ n=0 I n=0 n=0 To conclude, we use the fact that n EB(n) is decreasing and that 0 > EB(n) > —Cn7^5 (we do not need a sharp lower bound here). Then EB(N) < y II^II^bW < E \\*n\\2EB(n) n<2N n<2N oo = y\\*n\\2EB(n)- y \\*n\\2EB(n)2N n>2N The error term with n > 2N > (tf, M^) + X4/5 can be estimated by the variance bound and the Cauchy-Schwarz inequality £ n*„ii v < e (^«) n>2N n>2N / oo 3/5 7/10 < En*. |2 2 n Kn=0 3/10 ■2lT,n7/10 = «^,A/"2^)) AT6/5 3/10 7 < C(JV: 2\7/10 AT6/5 3/10 Thus we conclude that EB(N)