Mathematical Quantum Mechanics II Phan Thanh Nam Summer 2020 Contents 1 Principles of Quantum Mechanics 4 1.1 Hilbert spaces................................... 4 1.2 Operators on Hilbert spaces........................... 5 1.3 Principles of Quantum Mechanics........................ 7 1.4 Many-body quantum mechanics......................... 9 2 Schrodinger operators 15 2.1 Weyl's theory................................... 16 2.2 Min-max principle ................................ 17 2.3 Sobolev inequalities................................ 18 2.4 IMS formula.................................... 21 2.5 Schrodinger operators with trapping potentials................. 22 2.6 Schrodinger operators with vanishing potentials................ 25 2.7 HVZ theorem................................... 28 2.8 How many electrons that a nucleus can bind? ................. 34 1 2 CONTENTS 3 Hartree theory 40 3.1 Existence of minimizers: trapping potentials.................. 41 3.2 Existence of minimizers: vanishing potentials.................. 45 3.3 Existence of minimizers: translation-invariant case............... 56 3.4 Hartree equation................................. 64 3.5 Regularity of minimizers............................. 65 3.6 Positivity of minimizers.............................. 68 3.7 Uniqueness of minimizers............................. 75 3.8 Hartree theory with Dirac-delta interaction................... 79 4 Validity of Hartree approximation 81 4.1 Reduced density matrices ............................ 82 4.2 Hoffmann-Ostenhof inequality.......................... 85 4.3 Onsager's lemma................................. 88 4.4 Convergence to Hartree energy.......................... 90 4.5 Convergence to Hartree minimizer........................ 96 4.6 Short-range interactions ............................. 106 5 Fock space formalism 114 5.1 Creation and annihilation operators....................... 115 5.2 Second quantization ............................... 119 5.3 Generalized one-body density matrices..................... 123 5.4 Coherent/Gaussian/Quasi-free states...................... 128 6 Bogoliubov theory 137 6.1 Bogoliubov heuristic argument.......................... 137 6.2 Example for the homogeneous gas........................ 142 6.3 Bogoliubov transformation............................ 145 6.4 Diagonalization of block operators........................ 158 6.5 Characterization of quasi-free states....................... 165 6.6 Diagonalization of quadratic Hamiltonians................... 174 7 Validity of Bogoliubov approximation 182 7.1 Bogoliubov Hamiltonian............................. 183 7.2 Unitary implementing c-number substitution.................. 186 7.3 Transformed operator............................... 191 CONTENTS 3 7.4 Operator bounds on truncated Fock space................... 194 7.5 Improved condensation.............................. 199 7.6 Derivation of Bogoliubov excitation spectrum ................. 205 7.7 Extension to singular interaction potentials................... 212 Chapter 1 Principles of Quantum Mechanics 1.1 Hilbert spaces Definition (Hilbert Spaces). A space Jrf? is a Hilbert space if • ffl is a complex vector space; • it is equipped with an inner product (•, •) which is linear in the second argument and anti-linear in the first {x,\y) = \{x,y), {Xx,y) = \{x,y); II • II) is a Banach (complete normed) space with norm \\x\\ = \J(x, x). A Hilbert space J^f is separable if there exists a finite or countable family of vectors {"ura}ra>i which forms an orthonormal basis. In this case, we can write x = n>l 'y^(un, x)un, Vi G . Consequently, we have Parsevel's identity Ix||2 = n>l We will always work with separable Hilbert spaces. 1.2. OPERATORS ON HILBERT SPACES Review: Riesz (representation)/Banach-Alaoglu/Banach-Steinhaus theorem. 1.2 Operators on Hilbert spaces -\ Definition (Operators on Hilbert Spaces). By an operator A on Jrf? we mean a linear map A : D(A) —> Jrf? with a dense, subspace D(A) (domain of A). The adjoint operator A* is defined by D(A*) = jx G 3tf \3A*x G 3tf : {x,Ay) = {A*x,y), My G 0(A)}. The operator A is self-adjoint if A = A*. The concept of self-adjointness is very important in quantum mechanics. Mathematically, it enables various rigorous computations, thanks to the Spectral theorem. / Theorem (Spectral theorem). Assume that A is a self-adjoint operator on a separable Hilbert space ,3^. Then there exists a measure space a real-valued measurable function a : Q —> M. and a unitary transformation U : J4f —> L2(tt) such that UAU* = Ma. Here Ma is the multiplication operator on L2(tt), defined by (MJ)(x) = a(x)f(x), D(Ma) = {/ G L2(Q), af G L2(Q)}. We can choose = a (A) xtfcK2 and a(X, n) = X. In practice, the self-adjointness is not always easy to prove. It is however easier to check whenever an operator A is symmetric, namely (x,Ay) = (Ax,y), Vx, y G D(A). 6 CHAPTER 1. PRINCIPLES OF QUANTUM MECHANICS Exercise. Prove that the fallowings are equivalent: 1. A is a symmetric operator; 2. {x, Ax) G R for all x e D(A). 3. A* is an extension of A, namely D(A) C D(A*). Thus 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 Friedrichs' extension and Kato-Rellich theorem. 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 AF by Friedrichs' method. The Friedrichs extension preserves the ground state energy inf {x,Ax)= inf (x,Apx). x£D(A),\\x\\=l x£D(AF),\\x\\=l The quadratic form domain Q(Ap) is the same with Q(A). Recall that we define Q(A) as the closure of D(A) under the quadratic form norm ||:e||q(a) = ^{x, (A + C + l)x). However, the domain D(Ap) is often not known explicitly. For the latter issue, the Kato-Rellich theorem gives a better information on the domain of the extension. /-\ Theorem (Kato-Rellich theorem). Assume that we can write A = Aq + B, where Aq is self-adjoint and B is a small perturbation of A$, in the meaning that \\Bx\\ < (1 - e)\\A0x\\ + C£\\x\\, Vx e D(A0) c D(B), for some constant e > 0 independent of x (we say that B is Ao-relatively bounded with the relative bound 1 — e). Then A can be extended to be a self-adjoint operator on the same domain of A0. Review: Bounded/compact/Hilbert-Schmidt/trace class operators. 1.3. PRINCIPLES OF QUANTUM MECHANICS 1.3 Principles of Quantum Mechanics 7 Definition (Principles of Quantum Mechanics). A quantum system can be described by a (separable) Hilbert space . • A pure state is a rank-one projection \x){x\ with a normalized vector x 6 ,3^ (we use the bra-ket notation). A mixed state is a trace class operator T on Jrf? such that r = r* > 0, Tr r = 1. By Spectral theorem any mixed state is a super-position of pure states, namely where {xn} is an orthonormal family in Jrf? and ^n > 0, Y2n£n = 1- • The Hamiltonian H is a self-adjoint operator on Jrf? which corresponds to the energy {x, Hx) or Tr(HT). The ground state energy is If the infimum exists, then the ground state solves the Schrddinger equation Hx = Eqx. Other elements of the spectrum cr(H) corresponds to excited states. • At a positive temperature T > 0, the minimizer of the free energy is given uniquely by the Gibbs state Tt = 1e~H^T, Zt = Tr(e~H/T) (if the partition function Zt is finite). • The evolution of the quantum system is determined by the time-dependent Schrodinger equation x(t) = e~ltHx0. n>l Eq := inf (t(H) = inf {x, Hx) inf Tr(HT r>o,Trr=i J CHAPTER 1. PRINCIPLES OF QUANTUM MECHANICS Exercise. Prove that if the infimum E0 := ioia(H) = inf (x,Hx) = inf Tt(HT). \\x\\=i r>o,Trr=i is attained for a mixed state T, then it is also attained for a pure state \x){x\. Exercise. Prove that any ground state \x){x\ satisfies the Schrddinger equation Hx = Eqx. Hint: For any y 6 // . define x£ = (x + ey)/\x + ey\\. Then the functional e i—> {x£, Hx£) has a local minimum at e = 0. / Exercise. Assume that the partition function is finite for some temperature Tq > 0. 1. Prove that ZT it is finite for all T 6 (0, To). This implies that the Gibbs states is well-defined for all T 6 (0, T0). 2. Prove that the free energy is finite for all T 6 (0, T0) and lim ET = Eq (the ground state energy). Hint: You can use the Gibbs variational principle ET = —T\ogZT. 1.4. MANY-BODY QUANTUM MECHANICS 1.4 Many-body quantum mechanics -\ Definition (Tensor product). Let J^f and Jrf?2 are two Hilbert spaces. The tensor product space J^f ® J/?2 is a Hilbert space ^x®^2 = Span{u ®v\u^ J^i, v G J^2}. Here the closure is taken under the norm of .3%[® which is given by the inner product (u1®u2,v1®v2)jfl®M = {u1,v1)^1{u2,v2)^2. Let A\ and A2 are operators on fflx and Jrf?2, respectively. Then the tensor product operator Ax ® A2 is an operator on fflx ® ,y^2 defined by Ax ® A2{ux ® u2) = (Axux) ® (A2u2), D(Ax ® A2) = D(Ax) ® D(A2). More generally, we can define the tensor product space fflx ® ^2 <8> ... <8> -J^n and the tensor product operator Ax® A2® ... ® A^. In particular, if Jtf[ = .3^2 = ... = J^at, then we write Jffx®Jffx® - ®^x = J?xm ■ Remarks: • The tensor product J^i ® is different from the direct product J^i x J^. In particular, for any A G C we have (A-ui) ®u2 = \{ux ® u2) = ux® (\u2) and similarly (XAx) ®A2 = X(Ax ® A2) = Ax® (XA2). • The notation J^i ® J^f2 ® ... ® J^at is consistent thanks to the Associative Property (8> ^r2) ® j% = ® ® The same applies to the tensor product operator Ax® A2® ... ® AN. 10 CHAPTER 1. PRINCIPLES OF QUANTUM MECHANICS f Exercise. Assume that {un}n>i is an orthonornal basis for and {vm}m>i is an orthonornal basis for J^. Prove that {un®vm}m:n>i is an orthonornal basis for Jrf?i®J#2- Exercise. Let ,y^\ and J#2 be two Hilbert spaces. Assume that the sequence {un}n>i is bounded in J^i and the sequence {vn}n>i converges weakly to 0 in J#2- Prove that un ® vn —^ 0 weakly in J^i <8> J^. ^ Exe Exercise. Prove that for any d,N > 1, we have L2(RdN) = L2(Rd)®N. f \ Definition (Many body quantum systems). Consider a quantum system of N particles, where the i-th particle is described by the Hilbert space and the Hamiltonian hl. Moreover, assume that the interaction between the i-th and j-th particles is described by an operator Wl3 on <8> J^. Then the combined system of N particles is described by the interacting Hamiltionian n HN = Y,K+ W*r i=l 1... <8> hi ® ... ® 1 (the identity 1 is put everywhere except the i-th position). The same applies to WlJ} for example W\i is identified to W\i®\_^?>®...®\ Remarks: The above expression is a bit formal as we did not specify the domain of relevant operators. In practice, we will consider the case where Wl3 is relatively bounded with respect to hi + hj, with an arbitrary small relative bound. In this case, by the Kato-Rellich theorem the interacting Hamiltonian is self-adjoint on the same domain with the non interacting Hamiltionian H°N = h1 + ... + hN. 1.4. MANY-BODY QUANTUM MECHANICS 11 f Exercise (Non-interacting Hamiltonian). Assume that for any i = 1,2,...,N, the Hamiltonian ht is self-adjoint on J^f. Consider the N-particle system with the non-interacting Hamiltonian H% = hi + ... + on ® ... ® J^at. 1. Prove that H% is self-adjoint with the domain D(H%) = Dih^ D(h2) ... D(h N Here the closure is taken with the operator norm \\^>n\\hon = \\^n\\ + ll-^Ar^wll-2. Prove that the ground state energy of H^ is N mfa(H^) = inf a (hi) (both sides can be —oo). i=i 3. Prove that if ul is a ground state of hi, then ui ® ... ® uN is a ground state of H%. In practice, we will mostly consider identical particles. For instance, every electron in the universe has the same mass, electric charge and spin. To work with identical particles, we will always assume that the corresponding one-body operators hi is the same for all i, and that the interaction operator Wl3 is the same for all i and j (in particular, Wl3 = Wji). The notations h% and W%] are still useful to indicate which particles that the operators act. Then the iV-body Hamiltonian N HN = ^2ht+ Yl W*r i=l l ,yf®N by LSa(u\ ®U2® ... <3 UN) = -UCT(i) Ua(2) <8> ... <8> Ua(Ny • For N identical bosons, the corresponding Hilbert space ,y^®aN is the symmetric 12 CHAPTER 1. PRINCIPLES OF QUANTUM MECHANICS subspace of J{?®N, namely Ua(VN) = VN, G je®°N, Va G SN. • For N identical fermions, the corresponding Hilbert space ,yf®aN is the antisymmetric subspace of Ji?®N, namely Ua{^N) = sign(a)^v, v^at G J>ff®aN, Va G SN. The latter identity is called Pauli's exclusion principle. Exercise. 1. Prove that the operator Ua defined as above is a unitary transformation. 2. Prove that the operators P+ = {Niy1 Ua, P- = (AH)"1 sign(t7)£7ff are orthogonal projections, namely P± = P± = P±. 3. Prove that = P+{,^N) and = P_{J>F®N). Exercise. Assume that {un}n>i is an orthonormal basis for Jff. Prove that {P±(un ul2 ... iiliV)}ll,...,liV>i is an orthogonal basis for J>ff®a/aN. The simplest example for a bosonic state is the Hartree state (pure tensor product state) u®N(xi,xn) = u(xi)...u(xn) where u is a normalized vector in . The simplest example for a fermionic state is the Slater determinant (ui a u2 a ... a uN)(xi,xN) = —= det {ui(xj))i R an external potential; • W = W(—.) : Rd —> R an interaction potential (it is even, hence W%] = WJ%). Recall that the bosonic space L2{Rd)®aN contains all symmetric functions, namely ^jvO^I) ■ ■■,xl, ...,x3,xN) = tyN(xi,Xj, xl,xN), Vz 7^ j while the fermionic space L2(Rd)®aN contains all anti-symmetric functions ^jvO^I) •••) xn •••) xji •••) xn) = ~^n(x1, Xj, xl,xN), Vz 7^ j. In this chapter the particle statistics does not play an important role, so at first reading you may think of Hn acting on the full space L2(RdN) for simplicity We will study some general spectral properties of the Schrodinger operators. We will always 15 16 CHAPTER 2. SCHRÖDINGER OPERATORS assume that the interaction potential W is relatively bounded with respect to —ARd. For the external potential, we distinguish two different cases: • The trapping case Vix) —> +00 as |x| —> 00; • The vanishing case V(x) —> 0 as |x| —> 00. The spectral properties of these two cases are very different. In the first case, the Hamiltonian HN has discrete spectrum with eigenvalues converging to infinity. This follows the same analysis that we have discussed in MQM1 (we will recall below). In the second case, the interaction operator is not a compact-perturbation of the kinetic operator, leading to a big change on the essential spectrum in comparison to the one-body Schrodinger operator. 2.1 Weyl's theory Let us quickly remind some important tools to study the Schrodinger operators. First we recall some general facts from spectral theory. \ Definition (Spectrum). Let A be a self-adjoint operator on a Hilbert space . Then its spectrum is (J{A) = {A G R : (A — A)-1 is a bounded operator }. The discrete spectrum adis(A) is the set of isolated eigenvalues with finite multiplicities. The essential spectrum is the complement aess(A) = a(A)\adis(A). f Exercise. Consider the multiplication operator Ma on L2(Q,yu) which is self-adjoint with the domain D(Ma) = {/ G L2 : af G L2}. Prove that • A G cr(Ma) iff yu(a_1(A — e, A + e)) > 0 for all e > 0, namely a(Ma) = ess-range(a). • A is an eigenvalue of Ma iff yu(a_1(A)) > 0. • A G Odis(^a) iff A is an isolated point of a(Ma) and 0 < yu(a_1(A)) < 00. 2.2. MIN-MAX PRINCIPLE 17 By the spectral theorem, any self-adjoint operator is unitarily equivalent to a multiplication operator. However, this abstract result is not very helpful in application, as it is hard to compute the measure \i. Here is a general characterization of the spectrum. Theorem (Weyl's Criterion). For any self-adjoint operator A on a Hilbert space ,3^: • 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. • A G aess(A) iff there exists a Weyl sequence {xn} C D(A) such that \\un\\ = 1, un —^ 0 weakly, ||(^4 — A)-u„|| —» 0 as n —> oo. In practice, Weyl's Criterion is very useful to study the essential spectrum. A famous consequence of Weyl's Criterion is Theorem (Compact perturbation does not change essential spectrum). Let A be a self-adjoint operator on a Hilbert space. Let B be a symmetric operator which is A-relatively compact, namely D(B) C D(A) and B(A + i)_1 is a compact operator. Then A + B is self-adjoint on D(A) and aess(A + B) = aess(A). Exercise. Prove the above corollary using Weyl's Criterion theorem. Hint: You can write B = B(A + *)_1(^ + *). 2.2 Min-max principle A useful tool to study the discrete spectrum below the essential spectrum is the min-max principle. ^Theorern^^^i^^axPrhicrple^^Le^^6^^^^ 18 CHAPTER 2. SCHRÖDINGER OPERATORS Assume that A is bounded from below and define the min-max values inf McD(A) sup {u, Au) . Nl=i dim M=n Then we have inf aess(A) = fj,oo(A) := lim fj,n(A). Moreover, if /in(A) < ^^(A), then /ii,.. . /in are the lowest eigenvalues of A. J Remarks: • In the above definition, the condition M C D(A) can be replaced byMcJ) for any subspace 2) which is dense in the quadratic form domain Q(A). Thus in practice, we can compute the min-max values even if we do not know the domain of A. For example, if A is the Friedrichs' extension of a (densely defined) operator Ao, then the min-max values can be computed using the domain of Ao. • It is obvious that /in(A) is an increasing sequence when n grows. Thus the limit Hoo{.A) := Hindoo /in(A) always exists, even it can be +oo. • If /100(A) = +00, then the strict inequality /in(A) < ^^(A) trivially holds for all n = 1,2, .... Consequently, all min-max values become eigenvalues and they converge to +00. In this case we say that A has compact resolvent because (A + C)_1 is a compact operator for any C > —/ii(A)). • The min-max values is monotone increasing in operator, namely if A < B, then In particular, if A < B and A has compact resolvent, then B has compact resolvent. fin{A) 1 and s > 0, define Hs(Rd) := {/ G L2(Rd) I |ifc|V(ifc) G L2(Rd)} rai/i / i/ie Fourier transform of f. This is a Hilbert space with the inner product if,9)H-= f(k)g(k)(l + \27rk\2)sdk Remarks: • We use the following convention of the Fourier transform f(k) = [ < • On the Sobolev space Hs(M>d), we can define the weak derivative via the Fourier transform D^f(k) = (-2nik)af(k) which belongs to L2(M.d) for any multiple index a = (a±,a.a) with \a\ = ai + .-.+a^ < s. • In the above definition and the Sobolev inequalities below, the power s is not necessarily an integer. 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 LP(Rd) < C||/||ff»(Kd), V/ G Hs(Rd) 2 d/2 20 CHAPTER 2. SCHRÖDINGER OPERATORS We say that Hs(M.d) c Lp(M.d) with continuous embedding. When s > d/2 we also have the continuous embedding Hs(M.d) c ^(ird) (the space of continuous functions with sup-norm). Remarks: • 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 lLPW)<^ll(-A)s/2/|U (on the right side we do not put the full norm of hs, but only the seminorm of hs). • In principle, for any given power s > 0, the Sobolev inequality becomes weaker when the dimension d grows. For example, h\r) a l2(r)r]^(r), h1^2) c P| lp(r2), h1^3) c Pi Lp(R2). 2l and s > 0. Then for any bounded set c rd, the operator tQ : Hs(rd) ->■ Lp(Md) is a compact operator, where 2 d/2. When s > d/2, we also have the compact embedding 1q : Hs(m.d) —> ^(lRd). Remark: An easy way to remember the Sobolev compact embedding is that if un —^ 0 weakly in any Sobolev space i/s(lRd) with s > 0, then for any R > 0 we have ||li„l(|x| < i?)||L2(Rd) -)■ 0. 2.4. IMS FORMULA 21 Then the strong convergence in LP follows by a standard interpolation (Holder inequality); this is the reason we have to avoid the critical power (end-point). Exercise. Assume that un —^ 0 weakly in a Sobolev space Hs(Rd) with s > 0. Prove that up to a subsequence n —> oo, we can choose Rn —> oo such that \\unl(\x\ < -Rn)||i2(Rd) -)■ 0. Is it really necessary to take a subsequence? 2.4 IMS formula Another helpful result from real analysis is the IMS formula, named after Ismagilov, Morgan, Simon and Israel Michael Sigal. This provides with a localization technique in the position/configuration space. Theorem (IMS formula). For any smooth function ip : Rd —> R (e.g. <^71 or Lipschitz), we have y2(-ARd) + (~ARrf)y2 f A \ IV7 12 Consequently, if smooth functions {Vj}j=i form a partition of unity, k 3=1 then k k -ARd = ^<^(-Ar*)Vj - |V^-|2. 3 = 1 3 = 1 22 CHAPTER 2. SCHRODINGER OPERATORS 2.5 Schrodinger operators with trapping potentials Theorem. Consider the Schrodinger operator N HN = ^(-AXt+V(xl)) + W(Xl-Xj) i=l KkjXN acting on L2(RdN) or L2(Rd)®°N or L2(Md)®aAr. Assume that • W(E Lp(Rd) + L°°(Rd) with p > max(d/2,1) • V E L^oc(Rd) and V(x) ->■ +oo as \x\ ->■ oo. Then HN, originally defined on the core domain of smooth functions with compact support, is bounded from below and can be extended to be a self-adjoint operator by Friedrichs method. Moreover, H^ has compact resolvent, namely it has discrete spectrum with eigenvalues converging to +oo. Proof. Step 1. We prove that Hn is bounded from below. Consider the external potential V : Rd —> R. For any e > 0 we can write V = V1 + V2, ||Vi||LP(Kd) < e, V2>-C£, lim V2{x) = +oo. | x\—>oo Consider a wave function \&jv £ C^°(RdN) (or a symmetric/anti-symmetric one) by Holder's inequality we have (VniV^x^Vn) = / V1(x1)\^N(x1,...,xN)\2dx1...dxN > Rd(iV- ( f |Vi(xi)|pdxiN) ( f |^at(xi, XN)\2qdxi\ dx2...dxN with 1+1=1. V Q Here we have i/p ( / \V1(x1)\pdx1) " =\\V1\ V had / LP < e. The condition p > max(l, d/2) implies that q < oo for all d > 1, and moreover q < Idj(d — 2) 2.5. SCHRÖDINGER OPERATORS WITH TRAPPING POTENTIALS 23 in case d > 3. Thus by Sobolev's inequality for i/1(lRd), (/ \*N(x1,...,xN)\2«dx1)1/q < [ 1(1 — Axl)1/2^Ar|2dx1. In summary {*N, ViixjVv) >-Ce [ ( [ |(1 - Axl)1^N\2dx1)dx2...dxN = -Ce{*N, (1 - AX1)^). This bound can be written in the compact form V1(x1) > -Ce(l-Axi). Similar estimates holds for Vi(x2), ...,Vi(xn). Thus n n n n n i=i i=i i=i i=i i=i Consider the interaction potential W : Rd ->■ R. For any e > 0 we can write W = W1 + W2, ||VFi||LP(Kd) < e, ||W2||l°° < Ce Similarly as above, for any wave function £ C£°(M.dN) (or a symmetric/anti-symmetric one) by Holder's inequality and Sobolev inequality we can bound (v&jv, VFi(xi - x2)^N) = / W1{x1 - x2)\^N(xi, ...,xN)\2dx1...dxN JWLdN ( / - x2)\pdx1) ( / |^at(xi, ...,XAr)|2gdxi) dx2...dxf >- f ||VFi||LP(Kd)fc / |(l-AXl)1/2^Ar|2dx1)dx2...dxAr > -Ce{*N,(l-AXl)*N). Here again we use the notation ^ + ^ = 1 and the condition q < oo, q < 2d/{d — 2) for d > 3. The only difference to the previous treatment of the external potential is that we use the translation-invariance of the interaction potential and the Lebesgue measure which ensure that ( [ \W1(x1 - x2)\pdx1) /P = ||Wi||LP(ic«A < e. V ./rod / 24 CHAPTER 2. SCHRODINGER OPERATORS The above estimate also holds for W\{xl — xA for any i ^ j. Thus 1 -CNe^^N, (-Axi)*n) ~ Ce,N. i=i The potential W2 is bounded, and hence for the total interaction part, we have AT Y {*n, W(Xl - x3)*N) > -CNe Y(*n, (-A.J^at) - C£,N. lYV2(xl)-C(N + l)eY(-A^)-Ce,N. i=l 1 0 small enough such that C(N + l)e < 1/2. Thus N N HN = Y(-A*i + V(xi))+ Yl W{xl-xj)>YJ(-^JrV2{xl))-CN. 1 = 1 1Y(- 2A* + V^)) ~ Cn- i=i and the Min-max principle, it suffices to prove that the operator ~ N 1 Hn = Y{-2Ax' + V^Xi)) %=i 2.6. SCHRÖDINGER OPERATORS WITH VANISHING POTENTIALS 25 has compact resolvent. This operator can be written as 1 N HN = --AKdN + U(X), U(X) = ^2v2(xi), X=(x1,...,xN)£RdN. i=i If Hn is an operator on the full space L2(M>dN), then we can interpret Hn as a one-body Schrodinger operator on L2(M>dN). The condition lim^i^oo V^ix) —> +oo implies that lim U(X) = +oo. Therefore, —|ARdiv + U(X) has compact resolvent (we have proved this in MQM1). Consequently, the original operator HN has compact resolvent. Now consider the case when HN is an operator on the bosonic space. Then by the definition, the min-max value of Hn is fin(HN) = inf sup {tyN, HN^N) ■ \\u\\ = l Here the infimum is taken over all symmetric subspaces M of C^(M.dN). The infimum does not increase if we ignore the symmetry condition on M, namely the min-max values of the bosonic Hamiltonian Hn are bigger than or equal to those of the Hamiltonian on the full space L2(M>dN). Thus the bosonic operator Hn has compact resolvent. Similarly, the fermionic operator Hn also has compact resolvent. q.e.d. 2.6 Schrodinger operators with vanishing potentials Now we turn to the case when the external potential vanishes at infinity. A motivating example is the Atomic Hamiltonian with the Coulomb potentials W(x) = and V(x) = -Z\x\~x, x G R3. We start with the self-adjointness of the many-body Hamiltonian for general potentials. Theorem (Kato theorem). Consider the Schrodinger operator n HN = ^2(-AXt+V(xl))+ Y W& i=l KkjXN x3) 26 CHAPTER 2. SCHRODINGER OPERATORS acting on L2(RdN) or L2(Rd)®°N or L2(Md)®aAr. Assume that W,V G Lp(Rd) + L°°(Md), p > max(d/2,1) T/iera i/^r is self-adjoint operator with the quadratic form domain Q{HN) = H\RdN) or Hl(RdN) = P+H1(RdN) or H*(RdN) = P_H\RdN). Moreover, if we assume that W, V G Lp(Rd) + L°°(Md), p > max(d/2, 2) then the domain of HN is D{HN) = H2(RdN) or H2(RdN) = P+H2(RdN) or H2(R3N) = P_H2(RdN). Proof. Part 1. Consider the case W, V G Lp(Rd) + L°°(IRd),p > max(d/2,1). Proceeding exactly as in the case of trapping external potentials (now we just do not have V^x) —> oo as \x\ —> oo), then we obtain the lower bound 1 N Hn > ^ ^ ~Axi — CN- i=i Thus Hn is bounded from below. Consequently, it can be extended to be a self-adjoint operator by Friedichs' method. Moreover, by the same argument we also get the upper bound n Hn < 2 ^ — AXi + CN- i=i Thus the quadratic form domain of Hn is the same with the non-interacting Hamiltonian n — ax. = — ARdN, 1=1 namely Q(HN) = H\RdN) or Hl(RdN) = P+H\RdN) or H*(RdN) = P-H1(RdN). 2.6. SCHRÖDINGER OPERATORS WITH VANISHING POTENTIALS 27 Part 2. Consider the case W, V E Lp(Rd) + L°°(Rd), p > max(d/2, 2). We can prove that the external and interaction potentials are relatively bounded with respect to the kinetic operator. Let us consider the interaction potential W : Rd —> R. For any e > 0 we can write W = Wx + W2, ||VFi||LP(Kd) < e, \\W2\\L°° < Ce. For any wave function by Holder and Sobolev inequalities (for H2(Rdj) we can bound ||VFi(xi - x2)^at||2 = / - a;2)|2|^Ar(xi,xN)\2dx1...dxI JWLdN 'N <[ (f \w1(x1-x2)\pdx1)2/P( [ l^vOn,...,^)!9 JRd(N-l) \JRd / \JRd <[ \\Wi\\2LPm(c f 1(1-AXl)^Ar|2dx1)dr2...dr JWLd(N-l) V JRd / max(d/2, 2) implies that q < oo for all d > 1, and moreover g < Id I (d — A) if d > 4, allowing to use the Sobolev inequality. The above estimate also holds for W\{x% — Xj) for any i ^ j. Thus N Y \\Wi(xi - xj)*n\\ 0 small enough such that N I N i=l l 1) G L3"£(M3) + L3+£(M3), Ve > 0. Thus the condition LP + L°° in the above Theorem is clearly satisfied. • For Coulomb potential, instead of using Sobolev inequality you may also use Hardy's inequality -A > -ji- on L2(R3). ~ 4|x|2 v ' • The self-adjointness of the Atomic Hamiltonian was first proved by Kato in 1951. There is a nice story behind his proof; see "Tosio Kato's Work on Non-Relativistic Quantum Mechanics" by Barry Simon https://arxiv.org/pdf/1711.00528.pdf. 2.7 HVZ theorem Unlike the case of trapping external potentials, the Hamiltonian with a vanishing external potential has continuous spectrum. If the interaction potential is positive, the essential spectrum was determined by Huntiker, Van Winter and Zhislin in 1960s. Theorem (HVZ theorem). Consider the Schródinger operat or N HN = Y(-AXt+V(xl))+ Y W& i=l l 0 and W, V G Lp(Rd) + Lq(Rd), oo > p, q > max(d/2, 2). Then o-ess{HN) = [EN_X, oo), EN_1 = m{a{HN_1). Remarks: • The condition W > 0 is needed for the inclusion o-ess(HN) C [EN_1, oo). The inclusion [£W-i, oo) C aess(H^) always holds true without the condition W > 0. • For the one-body operator —A + U(x) on Rd, if [/ vanishes at infinity then it is a compact perturbation of the free Schrodinger operator —A. Therefore, by Weyl's theorem, we know that aess(-A + U) = a(-A) = [0, oo). The picture changes completely for the many-body Hamiltonian H^. The reason is that the interaction W[x% — Xj) does not vanish at infinity even if the function W : M.d —> M. vanishes at infinity (because xl and x3 may converge to infinity while their distance remains bounded). • The difference E^ — En_i (so called binding energy) is the energy needed to remove one particle from the bound state of a system of iV particles. The HVZ theorem tells us that En < -EW-i, and if E^ < En_i then H^ has a ground state (by the Min-max principle). Proof of HVZ theorem. We will use Weyl's criterion theorem. For simplicity we consider the case when HN acts on the full space L2(M>dN); the bosonic and fermionic cases follow small modifications. Step 1. We prove that [-E^-i, oo) C aess(HN). In this step we do not need W > 0. We take A ^ 0 and prove that Eat_i + A G (HN). By Weyl's theorem, we need to find a Weyl sequence {^^}n>i C L2(R.dN) such that 11^11 = 1, vff^O, \\(HN — EN_i — \)^^\\ —» 0 as n^oo. 30 CHAPTER 2. SCHRÖDINGER OPERATORS The choice of the Weyl sequence: • Since -EW-i = inf ct(Hn-i), we have .EW-i G a(H^-i) (the spectrum is closed). Therefore, by Weyl's theorem, there exists a Weyl sequence {^jv-i}n>i ^ L2(M.d(N~ls>) such that 11^11 = 1, - En-JV^W ->0 as n^oo. • Since A > 0, we have A G <7ess(—ARd). Therefore, by Weyl's theorem, there exists a sequence {u^} C L2(IRd) such that ||ii(n)|| = l, ii(n)^0, ||(-ARd - A)ii(n)|| ->■ 0 as n->oo. • Then we can choose namely It remains to check that }™>i is a desired Weyl's sequence for H^. It is actually correct. However, to make the proof easier, let us refine the choice of and a bit: by a standard density argument we can choose such that supply-1 C BKdN(0, Rn), suppii(n) C {x G Rd : 2Rn < \x\ < 3Rn} for some sequence Rn > 4f?„_i (you should check why?). Now let us prove that v&j^ = ® is a good choice. First, II^tv II — II^jv-lllll" II — l- Moreover, since the functions {u^} have disjoint supports, the functions {^^}n>i also have disjoint supports. In particular, {^$}n>i is an orthonormal family, and hence -± 0 weakly in L2(RdN). 2.7. HVZ THEOREM 31 Next, we decompose N-l HN = HN_ľ + {-AXN) + V{xN) + w(x* ~ xn)- i=i Therefore, by the triangle inequality \\{HN-EN^-\)^\\ < \\{HN^-EN^™\\ + \\{ a. N-l + \\V{xN)*%\ + Yi\\Wixi xN)^ 1=1 We have \\{HN_X-EN_^ ||(-AXJV-A)* \\V(xN)V \\W(xí-xn)^ H^JIK-A-A)^!! ^o, ||^-illll^H|| = \\V{x)t{\x\ > 2Rn)u^\\ -> 0 \\W(xí -xN)l(\Xi -xN\ > Rn)¥^_lU^\\ 0. For the last two convergences are obvious if we know that V(x) —> 0, W(x) —> 0 as |x| —> oo. More generally, if V, W G LP + Lq, then we can use Holder and Sobolev inequalities as in the proof of the self-adjointness, plus the fact that > 2Rn)\\ Here is bounded in H2(JLd). This concludes the proof of [En_i, oo) C aess(H^). Step 2. Now we prove that aess(H^) C [-E^v-i, oo). In this step we need W > 0. Take A + .EW-i G aess(H^). We prove that A > 0. By Weyl's theorem, there exists a Weyl sequence {^j^i C L2(RdN) such that From the above properties, we find that {w^ } is bounded in H2(M.dN), and hence the weak convergence in L2 can be upgraded to v&j^ —^ 0 in H2(M.dN) (see the exercise below). By Sobolev compact embedding theorem, we find that up to a subsequence as n —> oo, we can 0, \\{HN - EN_ľ - A)^|| ->■ 0 as n^oo. 32 CHAPTER 2. SCHRÖDINGER OPERATORS find Rr, —> oo such that (™) Physically, the latter convergence shows that v&j^ is not localized, namely at least one of N particles must escape to infinity. To trace the behavior at infinity, we use the IMS localization technique. • We choose two smooth functions x, ?] : IRd —»IR such that C X2+v2 = ^, suppx C {|x| < Rn}, suppT? C {|x| > Rn/2}, |VX| + |V7]| <— tin • On MdAr we have the partition of unity tRdN = ??2(xi) + x2{xi) = V2(%i) + rf{x2)X2{xi) + X2{x2)X2{xi) = V2(%i) + ri2{x2)x2{xi) + V2(x3)x2(x2)x2(xi) + ••• + X2{xn)-X2{xi) n Then we have \\7 0 and supp^o c BRdN(0, NRn), supp^ c {\xj\ > Rn/2} for all j > 1. Next we apply the IMS formula for ipo,ip^: n n n £, HN = Y1 VjHnVj - |V^|2 > (pjHjyipj - . 3=0 3=0 3=0 Therefore, by the choice of AT En-i + A = lim {^\HN^) > \immiYt{*%\ Ejv-iH^-^||2 + o(l)^oo. 2.7. HVZ THEOREM 33 For j = 0, using ip0HNip0 > ENipl and the fact \\^^ tB(o,Rn)\\ ->0we get {*%\ (poHNipoVff) > EN\\Vo¥p\\2 -> 0 as n oo. Therefore, we can also write For j = N we decompose AT HN = HN_t + (-AXN) + V(xN) + ^ W(Xl - xN) > EN_t + V(xN) i=i (here we use W > 0). Therefore, (pNHN(pN > EN-!(p2N - \V(xN)t(\xN\ > Rn/2)\. Thus (^\^HN^) > E^Wip^PW2 + (V™, 1^(^)1(1^1 > Rn/2)\^) >EAr_i||^Ar^)||2 + 0(1)^. Of course the same bound holds for j = 1, 2,N — 1 as well. In summary, we have proved that AT AT En-i + A > liminf V(^),^-^-^)) > liminf VE^W^f ||2 = n—¥oo —J n—¥oo —J 3=0 3=0 Thus A > 0. This ends the proof of aess(H^) C [£W-i, oo)- So far we have proved aess(H^) = [En_i, oo) when Hn acts on the full space L2(M.dN). When Hn acts on the bosonic/fermionic space L2(M.d)®s/aN we can proceed exactly the same, except that in the direction [-EW-i, oo) C aess(H^) we should choose the Weyl sequence as yff = P±*W x u<"> G L2(Rdf^N. q. e.d. 34 CHAPTER 2. SCHRODINGER OPERATORS Exercise. Let ,y^\ and J#2 be two Hilbert spaces such that ,3^2 C J^i and \\u\\m > IMI^b u ^ <^2- Assume that the sequence {un}n>i is bounded in J#2 and un —^ 0 in Prove that un —^ 0 in • 2.8 How many electrons that a nucleus can bind? Now we take a closer look at the Atomic Hamiltonian N 7 1 1=1 1 11 l 0 fixed at the origin in IR3. The particles interact via the Coulomb potential. Physically the Hamiltonian Hf}oin is an operator on the fermionic space L2(M?)®aN (electrons are fermions). Mathematically, we may also consider Hf}oin as an operator on L2(M?N) or L2(M.3)®3N. We will consider the nuclear charge Z > 0 as an arbitrary positive number, although it is an integer in practice. In this section, we address the following question: for a given nuclear charge Z, is there a ground state for HN1 i.e. "how many electron that a nucleus can bind?". From experimental chemistry, it is observed that a nucleus of charge Z can bind up to Z + 1 or Z + 2, but higher negative ions do not exist. Proving this fact rigorously for the Hamiltonian H^oia is a long-standing problem in mathematical physics, call the ionization conjecture. In the following, we will represent two fundamental results, one for the existence (Zhilin's theorem) and one for the non-existence (Lieb's theorem). Recall that from Kato's theorem, we know that H^oia is self-adjoint with domain H2(M:iN) r2a/s( or H2, (M3Ar), and that its ground state energy EN := mia{H\ atom\ N ) 2.8. HOW MANY ELECTRONS THAT A NUCLEUS CAN BIND? 35 is finite. Moreover, from the HVZ theorem we know that o-ess(H%°™) = [EN-1,oo) where EN_i is the ground state energy of Hf}™ (with the same nuclear charge Z). Consequently, H^OTa has a ground state if we have the strict binding inequality EN < EN_i. In principle, when EN = EN_i, H^ova may still have a ground state (although the ground state is very unstable as one particle can escape to infinity without losing any energy). Theorem (Zhilin's Existence Theorem). Consider H^oia as an operator on L2(M:iN) or L2(M:i)'^a/aN. For any 1 < N < Z + 1, we have the strict binding inequality En < En-i- Consequently, the Hamiltonian H^oin has a ground state. Proof. We will prove EN < EN_i by induction. This holds for N = 1 as E\ = — \ < 0 (the hydrogen atom). Assume that we have proved En-i < En_2 for some N < Z. Now we show that EN < EN_i. By the variational principle, we need to find a wave function ^N such that {^N,H^N) IR with supp<^ C {x G IR3 : 1 < |x| < 2} and IM|l2(r3) = 1- For any r > 0 we choose ^x) = iwAr)- Then supp<^ C {x G IR3 : r < \x\ < 2R}, ||<^||l2 = 1- • We choose the trial state 36 CHAPTER 2. SCHRÖDINGER OPERATORS Then II^AtIIl^RS") = ||^AT-i||l2(RM"-1))||^.r||.L2(]R3) = 1- It remains to show that (^N, H^ora^N) < E. N-l- Similarly to the proof of the HVZ theorem, we decompose 7 A?_1 1 HTm = HN™ +[-AXN- — + Y/T— 1 2=1 ' 1 Therefore, |2 H$™*N) = EN_X + || V^||2L2(R3) - / ^ Jw? i11 N-i 1 + i-||^jv-i(.Ti,...,a;jv-i)|2|vÄ(^jv)|2da;i-da;jv- i_1 Jm?N \xi ~ xn\ By the choice of ipn, we have II V^Ä||i-2(K3) = -^||Vv3||i2{r3) |x| R ./in3 1^1 Moreover, by Newton's theorem wWI»di= / l*«(*)l' „,< / / M£)fdT. |y-x| 7R3 max(|y|, |x|) 7R3 |x| 7R3 i?|x Consequently, V /-----|1'Ar_1(:r1, ...,xN_1)\2\cpR(xN)\2dx1...dxN Jw?N \xi ~ xn\ = / l^-lCxx,...,^!)^ / '/^^d^W. 0 sufficiently large we have 1 „ „9 N-l-Z f Mx)\2 , jfMlhm + —I Lj^-te < o. which implies that EN<{^N,H^N) 2Z + 1, then the Hamiltonian H^oia does not have a ground state on L2(R3N) ori2(R3)8«/siV. Proof. Assume that HN has a ground state ^N. Then it satisfies the Schrodinger equation Multiplying the Schrodinger equaiton with Ix^vI^tv and integrating we get 0 = {\xN\VN, (HN - EN)^N) = N-l | ^ If 7 N~X 1 = ( \xN\*N, H^ -EN-AXN--.-f + V 1- Xn\ . -, \Xi — Xtv 1=1 1 1 We have • (\xN\VN,(H%™£ - EN)VN) > 0. This follows from H%°% > EN_X > EN on the (N — l)-particle space, by the HVZ theorem. • {\xN\VN,-AXNVN) = %(vN, (\xN\(-AXN) + (-AXN)\xN\^N^ ^ 0 (the left side is real, why?). This follows from the IMS formula and Hardy's inequality Thus -A) x + x K-A) = ixm-AMxi1'2 - ivrixi1/2) 1 2bl1/2 V 4x2/ ~~ N-l r °^-z + E / 1=1 J^N \xn\ N-l \xi — Xtv 'AT 1=1 \Xn\ \Xi — Xtv 'AT Similarly we get for all j G {1,..., N} x 11 -I*, 12 - - / - o i ^7 N\ 2.8. HOW MANY ELECTRONS THAT A NUCLEUS CAN BIND? 39 Averaging over j 6 {1,..., N} we get N -'Villi N l i.e. JV < 2Z + 1. Here we have the strict inequality at the end because —■—J- > 1 for a.e. ...,x/v) G IR3Ar. g.e.d Remarks: • For Z = 1 (hydrogen atom), Lieb's theorem implies that the negative ion H does not exist. This is sharp because it is known (mathematically) that H~ exists. • For larger Z, the factor 2 in Lieb's bound is not sharp. For fermionic ground states, the above proof can be modified by multiplying the Schrodinger equation with Ixjv^^at instead of IxtvI^tv, leading to the non-existence when (my paper) N > 1.22Z + 3Z1/3. When Z —> oo, the non-existence of fermionic ground states is known when N > (l + e)Z for any e > 0. This so-called asymptotic neutrality was first proved by Lieb, Sigal, Simon, and Thirring (PRL 1984) and improved later in (CMP 1990a, CMP 1990b). The non-existence of fermionic ground states when N > Z + C for a universal constant C (possibly C = 2) remains an open problem. • When H^om acts on the full space L2(R3N) or the bosonic space L2(R3)®sN, a ground state exists up to N ~ \.2\Z. This was proved by Benguria and Lieb (PRL 1983). This is an evidence that the particle statistics changes dramatically the spectral properties of the quantum system. Chapter 3 Hartree theory Definition. Let v : rd ->■ r and w : rd ->■ R even. We define the Hartree functional En(u) := J (\Vu(x)\2 + v(x)\u(x)\2"\dx ^— // \u(x)\2\u(y)\2w(x — y)dxdy JwLd ^ ' 2 J 7RdxRd and the Hartree energy eH := inf{£H(«) = u G tf1^), |M|L2(Kd) = 1}. // a Hartree minimizer u0 exists, then it satisfies the Hartree equation I — A + v(x) + (w * \u0\ )(x) — jj^uq^x) = 0, x G rd for a constant ji G r (called the Lagrange multiplier or chemical potential/ The Hartree equation is also often called the Gross-Pitaaevskii equation or nonlinear Schrodinger equation, in particular when w = aSo (Dirac-delta distribution). In this case, the functional becomes £B(u):= (\Vu(x)\2+ V(x)\u(x)\2) +- // \u{x)\Adx JwLd \ J 2 J JRd and its minimizer satisfies ( - A + V{x) + a\u0(x)\2 - (i)uo(x) = 0, x G rd. 40 3.1. EXISTENCE OF MINIMIZERS: TRAPPING POTENTIALS 41 The Hartree/GP/NLS equation is an important topic in many areas of mathematics, e.g. nonlinear analysis, calculus of variations, and partial differential equations. In this chapter we study basic properties of Hartree theory and in the next chapter we discuss its connection to quantum Bose gases. Connection to many-body quantum mechanics. Consider a system of N identical bosons in Rd, described by the Hamilttonian N HN = ^2(-AXi+V(Xl))+ Yl w^~xo) i=l l■ R and W is even. Since the underlying Hilbert space is too large, it is often useful to restrict the consideration to the Hartree states V = u®N, \\u\\L2{Rd) = 1. The corresponding energy expectation (per particle) is exactly given by the Hartree functional — {u®N,HNu®N)= f (\Vu(x)\2+V(x)\u(x)\2)+- [[ \u{x)\2\u{y)\2w{x-y)dxdy =: £B(u) N JKd V J 2 J J with w = (N — 1)W. By the variational principle, the Hartree energy is always an upper bound to the ground state energy (per particle) of the full iV-body problem, namely EN := inf HNV) < iVeH. ll*lll,2(Rd)®siV=l The matching lower bound is nontrivial. We will prove that, under appropriate conditions on the potentials, EN = Neu + o(N). Moreover, we will prove that Hartree minimizers will give the leading order information to the ground states of the iV-body problem, leading to a rigorous justification of the Bose-Einstein condensation for some weakly interacting bosonic systems. 3.1 Existence of minimizers: trapping potentials 42 CHAPTER 3. HARTREE THEORY Theorem (Existence of Hartree minimizers: trapping case). Consider the Hartree functional £k(u) := / ('lV-u(x)l2 + l/(x)|-u(x)|2N)dx H— // \u(x)\2\u(y)\2w(x — y)dxdy with • we Lp(Rd) + L°°(Rd) with p > max(d/2,1), • V G Lfoc(IRd) and V{x) —> +oo as \x\ —> oo. Then the minimization problem eH := inf {sH(u) : u G tf1^), |M|L2(Rd) = l} has a minimizer (in particular en is finite). Proof. We use the direct method of Calculus of variations. Step 1 (Boundedness from below). Let u G H1(Md) with ||-u||L2(Rd) = 1. For any e > 0 we can write W = W1+W2, \\wxl\Lpmd) < e, ||w2||L°°(R|d** = f M*)I'(H • MaH*)d* < IIM^Iwlll^il * |m|2||lp < ||m|||2,||wi||lp||«|||2 with 1 1 - + - = 1. v q 3.1. EXISTENCE OF MINIMIZERS: TRAPPING POTENTIALS 43 The condition p > max(d/2,1) implies that 2q < 2* where 2d 2* = oo for d < 2, 2* =-- for d > 3. ~ ' d-2 By Sobolev inequality \\u\\2L2q -C£, lim V2(x) = +oo. | x\—>oo Using Holder and Sobolev inequalities we get |Ui(x)|Kx)|2dx < eC(||Vii|||2 + 1). By choosing e small enough, we find that \u(x)\2\u(y)\2\w(x - y)\dxdy + / |Vi(x)||ii(x)|2dr < -||Vit|||2 + C JWLd 2 for a constant C independent of u. Consequently, we get the lower bound ■1, -\Vu\2+ V2\ul2 C. Since U2 is bounded from below, we know that E^iu) is bounded from below uniformly in u. Thus en is finite. Step 2 (Minimizing sequence). Since en is finite, there exists a minimizing sequence {un}n>i C H1^) for eH, namely IWIlW) = 1, £n(un) eH as n -)■ oo. 44 CHAPTER 3. HARTREE THEORY From the above lower bound, we find that \Vun\2 + 2V2\un\2 is bounded. Thus {un} is bounded in the quadratic form domain of Q(—A + 2V2) = Q(—A + V2). By the Banach-Alaoglu theorem, up to a subsequence, we can assume that un —^ u0 weakly in Q(—A + V2). We will show that u0 is a minimizer for eH. Step 3 (Conservation of mass). The condition V2(x) —> +00 as |x| —> 00 implies that the operator —A + V2 has compact resolvent. Consequently, we have the compact embedding Q(—A + V2) C L2(IRd) (exercise). Thus the weak convergence un —^ uq in Q(—A + v2) implies the strong convergence un —> uq in L2(M.d). Therefore, uo\\l2(wld) — um ||un||L2(Rd) — 1- Step 4 (Semi-continuity). It remains to show that liminf £-a{un) > £n(u0). Since un is bounded in Q(—A + V2) C i/1^^) and un —> uq strongly in L2(lRd), by interpolation (Sobolev's and Holder inequalities) we find that un —> uq strongly in L9(lRd) for all 2 < q < 2* (2* = oo if d < 2 and 2* = 2d/(d - 2) if d > 3). Consequently, lim^ J J \un(x)\2\un(y)\2w(x - y)dxdy = J J \u0(x)\2\u0(y)\2w(x - y)dxdy. (see an exercise below). Similarly, for the external potential V = V\ + V2, using V\ g Lp(lRd) we have lim / Vi(x)\un(yx)\2dx = / Vi{x)\uQ{x)^dx (see an exercise below). Finally, since un —^ u$ weakly in the quadratic form domain Q(—A-V2), by Fatou's lemma for norms (see an exercise below) we have liminf /(|Vii„|2 + V2\un\2) > í(iViiol2 + V^2|^0|2)- n^oo J J In summary, eH = liminf£u{un) > £n{u0). 3.2. EXISTENCE OF MINIMIZERS: VANISHING POTENTIALS 45 This implies that u$ is a minimizer for en- q.e.d. Exercise. Let A be a positive self-adjoint operator on a Hilbert space Jrf? with compact resolvent. Prove that we have the compact embedding Q(A) C Jff. Hint: By Spectral Theorem you can write A = ^2n>1 \n\un){un\ with \n —> oo. The identity 1 : Q(A) —> Jrf? is a compact operator because it is the strong limit of finite-rank operators Bn = YZi=i \um){um\- Exercise. Let V g Lp(M>d) with p > max(d/2,1). Prove that if un —^ uq weakly in H1^), then lim / V(x)\un(x)|2dx = / V(x)\uq(x)\2Ax. Exercise. Let w g Lp(M>d) with p > max(d/2,1). Let {un}n>i, {vn}>i be bounded sequences in i/1(lRd) such that un —> uq strongly in L2(M.d) and vn —^ vq weakly in L2(Rd). Prove that lirn^ 11 \un(x)\2\vn(y)\2w(x - y)dxdy = I I \u0(x)\2\v0(y)\2w(x - y)dxdy. Exercise. (Fatou's lemma for norms) Assume that vn —^ v weakly in a Hilbert space. 1. Prove that liminf > \\v\\. 2. Prove that \\vn\\ —> \\v\\ if and only if vn —> v strongly. 3.2 Existence of minimizers: vanishing potentials Now we turn to the case when the external potential V vanishes at infinity. This case is significantly more difficult since some mass may escape to infinity leading to a possible lack of compactness. In fact, the existence of Hartree minimizers is not always guaranteed! We have to investigate all possibilities of losing mass at infinity. This is nicely done by the concentration-compactness method which has been developed since the 1980s by several people, including Lieb (Invent 1983) and Lions (AIHPC 1984a, AIHPC 1984b). 46 CHAPTER 3. HARTREE THEORY Theorem (Existence of Hartree minimizers: vanishing case). Consider the Hartree functional £^(u) := f (\Vu(x)\2+ V(x)\u(x)\2)dx+ - [[ \u(x)\2\u(y)\2w(x - y)dxdy with w, V g Lp(Rd) + Lp(Rd), max(d/2,1) < p, q < oo. For any a g [0,1] define 4(a) := inf {^(«) : u g H1^), \\u\\2L2{Rd) = \}. We denote by e^(a) the corresponding energy with V = 0 ("energy at infinity"). Then we always have the binding inequality 4(1) < 4(a)+ e°(l-a), vag[0,1] Moreover, if we have the strict binding inequality 4(i) <4(a)+ 4(i-a), vag[o,i), then the variational problem 4(1) ^as a fninimizer. In fact, for the existence of minimizers for 4(1)> we onty need the strict binding inequality when 4(a) has a minimizer. Remarks: • From the physical point of view, the binding inequality 4(1) < 4(a)+4(1 "a), vag[0,1] is rather obvious since the ground energy cannot be increased when we split the system into two parts: one with mass a staying bounded, and one with mass (1 — a) at infinity • The strict binding inequality 4(i) < 4(a)+ 4(i-a), vag[o,i) tells us that there is no possibility to put any positive mass at infinity (note that in the strict binding inequality we only requires a < 1). It is a nontrivial condition and depends heavily on the potentials V, w. 3.2. EXISTENCE OF MINIMIZERS: VANISHING POTENTIALS 47 • For repulsive interactions iw > 0), the energy at infinity is simply zero (see an exercise below). In this case, the binding inequality becomes 4« < 4(A), VAg[0,1] which is similar to the monotonicity EN < EN_i in the HVZ theorem (both always hold true). The strict binding inequality 4(i) < 4(a), vag[o,i) is thus similar to the binding condition EN < EN_i in the iV-body quantum problem. Proof. Step 1 (Boundedness from below). By the same analysis of the trapping case, we have / \V(x)\\u(x)\2dx + - ff \u(x)\2\u(y)\2\w(x-y)\dxdy <-\\Vu\\2L2 + C for all u g H1^) with |H|l2 < 1. Therefore, ^)>^||V^|||2-c. This implies that 4(A) is finite for every a g [0,1]. Step 2 (Binding inequality). Let a g [0,1). By a standard density argument, we can find a sequence {a„}„>i C H1(Md) such that supp(a„) C {|x| < n}, / \an\2 = a, £^(an) < e^a) + c^l)^^ ■JM.d (Explanation for the density argument: By the definition of 4(a), f°r any V > ® small we can find a function fv g H1(Md) such that ||/^|||2 = A and £^(fn) < 4(A) + V- Then since c^(1r3) is dense in H1(Md) and the mapping / 1—» £h(/) is continuous from H1(Md) to ir, we can replace fv by fv g c^qr3) with ||/^|||2 = A and £^(.fv) < 4(A) + 2??. By re-labeling 77 —> 0 by n —> 00, we get the sequence {an}.) Similarly, we can find a sequence {bn}n>i C H1^*) such that supp(fo„) C {|x| > 2n}, f \bn\2 = l-X, £^{bn) 2n\.) Now we define the trial state = an + bn, Mn > 1. Since an and bn have disjoint support, we find that \i C Hx{Kd) be a minimizing sequence for e^(l), namely IKIIlW) = 1, (Un) ~> e^(l). From the lower bound £h (Un) > 2-||Vii„|||2 - C we find that {un} is bounded in H1(Md). By the Banach-Alaoglu theorem, up to a subsequence, we can assume that un —^ uq weakly in i/1(lRd). Step 4 (Splitting of mass). Since un —^ u0 weakly in L2(lRd), Fatou's lemma tells us A := / |wo|2 < liminf / |itn|2 = 1. JRd n->oo JKd Moreover, if we denote vn ■= un - u0, 3.2. EXISTENCE OF MINIMIZERS: VANISHING POTENTIALS 49 then \vn\2 = \\un — u0\\2L2 = \\un\\2 + ||^o||2 — 23ft(-u„, u0) —> 1 + A — 2A = 1 — A. Step 5 (Splitting of energy). We prove that lim Ul{un) - £V(Uo) - £°K)) = 0. For the kinetic energy, since vn = un — uq —^ 0 weakly in i/1(lRd), we have ||Vii„|||2 - ||Vu0\\\2 + ||V7j„|||2 = 2K(Vii0, Vtj„) -> 0. For the external potential energy, V\ur, V\u0\ < \v\ K + vn\2 - \u0\2 \V\(\vn\2 + 2|-u„||-u0 < / \V\\Vr, W\\vn \V\\u0\2 ->■ 0. Here we used that J|V^||-uo|2 is finite because uq g H1(Md), and J|V^||ura|2 —> 0 because vn —^ 0 in H1(Md) (see a previous exercise). For the interaction energy, we have \un[x \ un |ii0(a:)|2|^o(y)|2 - \vn(x)\2\vn(y)\2 w(x - y)dxdy < \un(x)\2\un(y)\2 - |ii0(a:)|2|^o(y)|2 - \vn(x)\2\vn(y)\2 \w(x - y)|dxdy Writing un = uq + vn and expanding the difference \un(x)\2\un(y)\2 - \u0(x)\2\u0 \vn{xI vn (y)P we find several terms (whose absolute values) like \vn(x)\\fn(x)\\u0(y)\\gn(y)\, \vn(x)\\uo(x)\\fn(y)\\gn(y)\ where the functions fn,gn are bounded in H (R ). By the Cauchy-Schwarz inequality, we 50 CHAPTER 3. HARTREE THEORY can bound \vn{x)\\fn{x)I\u0(y)I\gn(y)\\w(x - y)\dxdy <(/ \vn(x)\2\u0(y)\2\w(x - y)\dxdy) i \fn(x)\2\gn(y)\2\w(x - y)\dxdy Here we used \fn{x)\2\gn{y)\2\w{x - y)\dxdy < C because fn, gn are bounded in i/1(IRd) (the interaction energy is bounded by the kinetic energy by Step 1) and \vn(x)\2\u0(y)\2\w(x - y)\dxdy ->■ 0 because vn —^ 0 weakly in H1(Md) and uq G H1(Md) (see a previous exercise). Moreover, by the Cauchy-Schwarz inequality again \vn{x)|\u0{x)I\fn(y)|\gn(y)\\w(x - y)\dxdy 1/2 < ( / \vn{.x)\\u0(x)\\fn{y)\ \w(x - y)\dxdy i ' \ 1/2 \vn{x)\\uQ{x)\\gn{y)f\w{x - y)\dxdy\ ->■ 0. Here we used that fn,gn are bounded in i/1(lRd) and l^'Wol1^2 0 strongly in L2(IRd) (see an exercise below). Thus in summary, for the interaction energy we obtain 0. \un(x)\ \un(y)\ -\u0(x)\ \u0(y)\ - \vn(x)\ \vn(y)\ wix - y)dxdy We conclude that lim Ul{un) - £^(u0) - £^(vn)) = 0. Step 6 (Conclusion from binding inequality). From the above estimates we find that e£(l) = lim £l(un) = £l(u0) + lim £^{vn) > e£(A) + e° (1 - A). On the other hand, we have the binding inequality e^(l) < e^(A) + e^(l — A). Thus here we 3.2. EXISTENCE OF MINIMIZERS: VANISHING POTENTIALS 51 must have that lim£°K) = e°(l-A), n—too namely vn is a minimizing sequence for e^(l — A), and ^K) = 4(A), namely u0 is a minimizer for 4(A)- In principle we only know that a < 1. Step 7 (Conclusion from strict binding inequality). If a < 1, we have 4(1) = 4(A)+4(1 "A) (and 4(a) has a minimizer). This violates the strict binding inequality. Putting differently if the strict binding inequality holds, then a = 1, and 4(1) nas a miminizer. q.e.d. -\ Exercise. Assume that /„ 0 m H^{Rd) and let g G H1^*). Prove that \fng\1/2 ->■ 0 strongly in Lp{Rd) for any 2 < p < 2* (with 2* = oo if d > 2 and 2* = ^ if d > 3). Exercise. Consider the Hartree functional £h (u) '■= / (Nu(x)\2 + V(x)\u(x)\2\dx + - // \u(x)\2\u(y)\2w(x - y)dxdy with V, w G Lp(Rd) + Lg(Rd) for some max(d/2,1) < p, q < oo. Let a G [0,1] and define the Hartree energy 4(a) := inf fan) I u G H\Rd), \\u\\2L2{Rd) = a}. 1. Prove that 4(A) < 4(A) < o. 2. Deduce that ifV,w > 0, then 4(A) = 4(A) = o. Here is an example of the application of the previous theorem. 52 CHAPTER 3. HARTREE THEORY Theorem (Existence of minimizers for bosonic atoms). Consider the Hartree functional for atoms £h(u):= [ (\Vu(x)\2-^-\u(x)\2)dx + - [[ H^)|2M^|2dxdy 7r3 V \x\ / 2 JJK3xK3 \x — y\ with Z > 0. Then for any 0 < A < Z, the variational problem E(Z, A) = inf {Sniu) \ u G H\R3), [ \u\2 = \\ Jr3 > has a minimizer. Proof. Since the Coulomb potential belongs to L3_£(1r3) + L3+£(1r3), we can apply the previous theorem. Here we are considering a positive interaction potential. Therefore, the binding inequality becomes E(Z, A) < E(Z, A'), v0 < A' < A and it suffices to show that when A < Z we have the strict binding inequality E(Z, A) < E(Z, A'), v0 < A' < A when E(Z, A') has a minimizer. It suffices to construct a trial state u such that / M2 < A, £n(u) 0 large there exists a function uR G ^(M3) such that supp(ufl) C {|x| < R}, J \uR\2 < A', £H(uR) < E(Z, A') + o^"1)^. 3.2. EXISTENCE OF MINIMIZERS: VANISHING POTENTIALS 53 Let uq be a minimizer for E(Z,\'). Let x, r\ : IR3 —> [0,1] be smooth functions such that x2 + v2 = 1) x{x) = 1 if \x\ < 1/2 and x{x) = 0 if |x| > 1. Define Xr{x) = x(x/R), uR = XrU0- Clearly we have supp(ii^) C {|x| < R}, J \uR\2 < J \u0\2 = A'. It remains to estimate the energy difference Su(ur) — £u(uo)- For the kinetic energy, by using the partition of unity Xr + v2r = i, xr(x) = x(x/R), and the IMS formula we can estimate J WuR\2- J\Vu0\2< J |V(X^o)|2 + J W(muo)\2- f |v^0|2 = / (|vx«|2 + |v^|2)KI2r/2 For the interaction energy, since the interaction potential is positive, we simply use to point-wise estimate |it^(x)| < |woOe)| to get 1 ff M-)l2My)l2dxdy_I ff K^I2l^)l2dxd, IR with supp-u C {x G IR3 : 1 < |x| < 2}, / Jr° For any R > 0 we choose \v\2 = e. 1 fx VRw = &r*v\R)- 54 CHAPTER 3. HARTREE THEORY Then suppvR C {x G R3 : R < \x\ < 2R}, / \vR\ Jm? 2=e. Step 3 (The trial state). We choose the trial state 0 small such that A' + e < A. Step 4 (Strict binding inequality). Now we estimate £h.{r) < 6h(A') which completes the proof. q.e.d Exercise. Consider the Hartree functional „ , s /" , x,9 k(^)l2\ , 1 /Y \u(x)\2\u(y)\2 , , As v |x| / 2 jJm.3Xm.3 f - yr parameters Z > 0 and 1 < s < 2. Prove that the minimization problem eH := inf {^(n) | u G tf1^), |M|L2(Kd) = l} has a minimizer. 56 CHAPTER 3. HARTREE THEORY 3.3 Existence of minimizers: translation-invariant case Now we consider the special case when the external potential is zero. This corresponds to the "problem at infinity". In this case, the Hartree functional Eft(u) := J |V-u(x)|2dx H— // \u(x)\2\u(y)\2w(x — y)dxdy JwLd 2 J 7RdxRd is translation-invariant, namely £&(u) =£h(u(- -y)), yu(EH\Rd), VyGMd. We know that if w > 0 (and w vanishes at infinity), then the corresponding energy 4(A) := inf\u G H1^*), \\u\\2L2 = A} is simply zero. However, if w < 0 (or if w has a non-trivial negative part), then in principle the energy e^(A) can be negative. Thus even if we start with a general (non-zero) external potential V, understanding the problem at infinity is still very helpful to justify the binding inequality Elil) < £g(\) + - A), VO < A < 1. On the other hand, the method in the previous section is not enough to deal with the translation-invariant case, because the binding inequality Elil) < £g(\) + - A), VO < A < 1. cannot hold true with V = 0 (just take A = 0). Therefore, we will need the following result. /-\ Theorem (Existence of Hartree minimizers: translation-invariant case). Consider the Hartree functional Eft(u) := J |V-u(x)|2dx H— // \u(x)\2\u(y)\2w(x — y)dxdy with w G Lp(Rd) + Lp(Rd), max(d/2,1) < p, q < oo. For any a G [0,1] define e° (a) := inf {£°(u) : u G H\Rd), \\u\\2L2{WLd) = a}. 3.3. EXISTENCE OF MINIMIZERS: TRANSLATION-INVARIANT CASE 57 Then we always have the binding inequality 4(A) < e°(Y) + e°(A- A'), VO < A'< A. Moreover, if we have the strict binding inequality 4(A) < 4(A7) + e° (A - A'), VO < A' < A, then 4(A) has a minimizer. In fact, for the existence of minimizers for e^(A), we only need the strict binding inequality when both 4 (A') and 4 (A — A') have minimizers. Remarks: • Note that in the above strict binding inequality we do not include the case A' = 0 (and the case A' = A). This is the main difference to the previous section . • Since 4(A') 5: 0 for all A' (see a previous exercise), the strict binding inequality in particular implies the non-vanishing condition 4(A) < 0. The main difficulty in the proof of the above Theorem is as follows: if u$ is a minimizer for 4(A), then un{x) = u0(x - yn), yn G Rd are also minimizers for 4(A)- On the other hand, if lim^oo \yn\ = +oo, then un —^ 0 weakly in iiT'-QR^). Similarly, there are several minimizing sequences for 4(A) that converge weakly to 0. Thus to apply the method of calculus of variations, we have to modify minimizing sequences using appropriate translations. Exercise. Assume that un —> u0 strongly in L2(Rd) Let {yn}n=i C Rd such that \yn\ +°o and denote vn{x) := un(x - yn), Vx G Rd, Vn G N. Prove that vn —^ 0 weakly in L2(Rd). The following key lemma provides a proper understanding of the vanishing case in the translation-invariant setting, namely the situation when we have the weak convergence to 0 up to all translations. 58 CHAPTER 3. HARTREE THEORY Lemma (Concentration-Compactness Lemma.). Let {un}n>i be a bounded sequence in H1^*). Then there are two alternatives: • Vanishing case: un —> 0 strongly in Lr(M.d) for all 2 < r < 2*, where 2* = oo if d < 2 and 2* = 2d/(d -2) tfd>3. • Non-vanishing case: There exist a subsequence {unk}k>i and a sequence {yk} C M.d such that Vk '■= — yk) converges weakly to a function vq ^ 0 in ^(W1). Proof. We define "the largest mass that stays in a bounded region" 9Jt({-u„}) := lim lim sup sup / ä-^oo eRd J\x. \ur,(x)\2dx. yGKd J\x-y\ 0. Then by the definition, there exists R > 0 such that lim sup sup / |itJl(x)|2dx > 0. ra->oo VGWLd J\x-v\oo yG]Rd J\x-y\ 0 such that unk(x)\2dx > e > 0, Vk > 1. '\x-Vk\ e > 0, Vk>l. |x|<ä On the other hand, H^i = ||-urafe||ffi is bounded. Therefore, up to a subsequence we can assume that vk —^ v0 weakly in i/1(lRd). By the Sobolev's embedding theorem we find that |x|<ä k^°°J\x\ e. Thus vq ^ 0, as desired. 3.3. EXISTENCE OF MINIMIZERS: TRANSLATION-INVARIANT CASE 59 Case 2: Vanishing: 9Jt({-u„}) = 0. Then for all R > 0 we have lim sup / dx = 0. ™^°°yeKd J\x-y\ 0 sufficiently large. We can write the space Rd as a union of finite balls Rd = |J B{z,R/2). zGZd Let x : Rd —> [0,1] be a smooth function such that x{x) \x\ > R. Define Xz(x) = x(x ~ z)- Then 1 if \x\ < R/2 and X{x) = 0 if 1 < ^(x)S - C' S IVXz(x)\2 2 and sufficiently close to 2, we can choose 4 0 = - - 1, g = 2 ' y 4-r < 2*, such that q(l - 0) = 2. The conditions q < 2* and g(l — 0) = 2 allow us to use Sobolev's inequality IXz^r, — \\Xzun\\2Lq 5: CllXz^nllfl"1- Thus in summary, for r > 2 and close to 2 we have \ r/2-1 I 12 \ II ||2 \Xzun\ i HXz^nllffl- Summing over z £Rd we obtain \Ur. zGlt < / ixz^r < c sup / iw ^d ly^ZdJWLd r/2-1 zGlt 60 CHAPTER 3. HARTREE THEORY \r/2-l ymd J\x-y\ 0 strongly in Lr(Rd) with r > 2 and close to 2. Since {un} is bounded in H1(Rd), by interpolation (Sobolev's and Holder's inequalities) we conclude that un —> 0 strongly in Lr(Rd) for any 2 < r < 2*. q.e.d. Exercise. Let {un}n>i be a bounded sequence in H1(Rd). Define 9Jt({-u„}) := lim lim sup sup / \u„ (x)\2dx yGKd J\x-y\i be a sequence of smooth functions : Rd —» [0, 1] satisfying sup X] \^(x)\2 + \VxXn(x) < oo. Prove that Y \\Xnu\\2m < C\\u\\2Hl{Rd), G H^R*). n>l The constant C > 0 is dependent on {Xn}n>i, but independent of u. Proof of the existence theorem. The finiteness of e^(\) and the binding inequality 4(A)<4(A') + e£(A-A'), V0\f |Vn|2-c Z JRd we find that any minimizing sequence for e^(A) is bounded in H1(md). Thus by the concentration-compactness lemma, there are two possibilities: vanishing case and non-vanishing case. Vanishing case: un —> 0 strongly in Lr(M.d) for all 2 < r < 2*. In this case, using w G Lp(Rd) + Lq(Rd) with p, q > max(d/2,1) we find that \un(x)\ \un(y)\ w(x - y)dxdy ->■ 0. cRd In fact, if w G Lp(lRd) for example, then by Holder and Young inequalities \un(x)\2\un(y)\2\w(x - y)\dxdy < \\\un\2\\LP>\\w * |ii„|2||lp < I Wl^'IMM Kill2 0. xRd Here 1 1 - + - = 1 p p and the condition p > max(d/2, 1) implies that 2 < p' < 2*. Consequently, we find that 4(A) = lim £°K) > 0. n—>oo However, it contradicts with the strict binding inequality (which particularly implies that 4(A) < 0). Non-vanishing case: Up to subsequences and translations, we can assume that un —^ uq weakly in i/1(lRd) with uq ^ 0. As proved in the previous section , we can split the energy lim (££K) - £&(uo) - £^(un - u0)) = 0. Denote A' := ||-uo|||2 > 0. Then \\un — uq\\L2 = \\un\\2L2 + H'Uoll!2 — 2'St(uniuq) —y A + A' — 2A' = A — A'. 62 CHAPTER 3. HARTREE THEORY Thus by the variational principle we have e°(A) = lim £°K) > £°M + lim £°(u„ - u0) > e£(A') + e° (A - A'). In comparison to the binding inequality 4(A)<4(A') + 4(A-V) we find that £^(uq) = e^(X') (i.e. u0 is a minimizer for e^(A')). Moreover, if the strict binding inequality holds, then we must have A' = A (as we have known already that A' = H^oll2^ > 0. Thus u0 is a minimizer for 4(^)- q.e.d. Here is an application of the above abstract theorem. Theorem (Choquard-Pekar Problem). Consider the Hartree functional with gravitational interaction potential J 2 JJK3xK3 \x — y\ Prove that for every a > 0, the minimization problem 4(a) := inf {£&(u) I u £ H1^*), \\u\\2L2 = a| Proof. Recall that — |x| 1 £ LA £(1r3) + L3+£(1r3). We need to check the strict binding-inequality 4(A) 0. In fact, take ip £ i/1(lR3) with ||y?|||2 = A. For every R > 0 define yR{x) = R-'^^x/R). Then ||<^i?|||2 = A and R ^RJJMßxM? \x-y\ 3.3. EXISTENCE OF MINIMIZERS: TRANSLATION-INVARIANT CASE 63 By taking R > 0 sufficiently large, we conclude by the variational principle 4(A) <0. Step 2: We prove that for all a > 0, for all 0 < 6 < 1, 4(ox) > #24(a)- Indeed, take a minimizing sequence {un}n>i C -/^(IR^) for e^(#a), i.e. |K|||2 = 6X, £°(un) ->■ e° (0A). Define — ^" ii ii2 — \ Then by the variational principle we have „0/x\ ^ c0/„^ _ illw.. 112 1 /7 K0*0|2K(y)|2 4(a) < £>„) = -||v^||i2 - — / / ' "V ' 7" drdy y 2^ J Jm3xM3 \x - y\ = G-^)l|v^ll|2 + ^K) Step 3: Using the estimates in Step 1 and Step 2, for every a > a' > 0 we can bound 4(a') + 4(a - a') = 4(yA) + 4(^a) >(y)4(a)+(^)4(a) = 6?(a). Thus the strict binding inequality holds, and hence 6q (a) has a minimizer for every a > 0. q.e.d. The above analysis can be also adapted to treat the Hartree problem with a general potential vanishing at infinity. 64 CHAPTER 3. HARTREE THEORY f Exercise (Choquard-Pekar Problem with an external potential). Consider the Hartree functional %{u) := !(\Vu\^V\uA-1- ff ^^dxdy J v > 2 JJE3xE3 \x-y\ with real-valued potentials V, w G Lp(M.d) + Lq(M.d), oo > p, q > max(d/2,1). Assume that V < 0 and V ^ 0. Prove that for every a > 0 the minimization problem e£(a) := inf {£%(u) | u G i^QR4*), |M||2 = a} has a minimizer. 3.4 Hartree equation Theorem (Hartree equation). Consider the Hartree functional £^(u) := I (\Vu(x)\2+ V(x)\u(x)\2)dx+ - 11 \u(x)\2\u(y)\2w(x - y)dxdy with V-,w G Lp(Rd) + L°°(Rd), V+ G Lploc{Rd) with p > max(d/2,1). Assume that for some a > 0 the minimization problem 4(a) := inf {s£(u) : u G H\Rd), \\u\\2L2{WLd) = x] has a minimizer u$. Then uq satisfies the Hartree equation ( - A + V(x) + (w* \u0\2)(x) - ii(X)^u0(x) = 0. in the distributional sense, namely VTp ■ V-uq + VTpu0 + Tp(w * |-u0|2)^o - filpuo =0, Vy? G C™(Rd). Here the constant ji(X) G R is called the Lagrange multiplier or chemical potential. Proof. Let us consider the case a = 1 for simplicity. By the variational principle, for every 3.5. REGULARITY OF MINIMIZERS 65

■ Vuq + VtpuQ + <^(k; * |-u0|2)-u0 — jitpuQ with „ = / (iv^)l' + v(l)K(*)|»W + // m*)|>Mi/)IM*- ./rod V / _/ _ I ]U> d ]U> d Replacing ^ by iy? (with i2 = —1) we find that 0 = 23 Vp> ■ Vuq + VtpuQ + ip(w * \uq\2)uq — jltpUQ Thus for all p) G C~(Md) we have Vp> ■ Vuq + F^-Uq + if(w * \uq\2)uq — IlipUQ = 0. q.e.d. 3.5 Regularity of minimizers Theorem (Hartree equation). Assume that uq G i/1(lRd) is a solution to the Hartree equation ( - A + V(x) + (w* \u0\2)(x) - ii^uo = 0. in the distributional sense. Assume that /i G R, V = V\ + V2 and • Vi, u> G Lp(Rd) + L°°(Kd) mfÄ p > max(d/2, 2); • 0 < V2 G L£c and \VV2(x)\ < C(F2(x) + 1). Then u0 G #2(Rd) and-Au0, Vu0 G L2(Rd), (w * \u0\2)u0 G L2(Rd). In particular, the Hartree equation holds in the pointwise sense. Remark: The conditions V2 > 0 and |\7V2(x)| < C(Vi(x) + 1) allow trapping potentials, e.g. 66 CHAPTER 3. HARTREE THEORY V2(x) —> +00 as |x| —> 00 and it does not grow faster than exponentially. Proof. Potentials vanishing at infinity. First let us consider the case V2 = 0, namely V = V\ vanishes at infinity. In this case, since V G LP + L°° with p > max(d/2,2), it is relatively bounded with the Laplacian with an arbitrarily relative bound, namely \\V0. Moreover, we have w * \u0\2 G L°°(lRd). Indeed, if w G LP for example, then by Young's and Sobolev's inequality we find that ||w*KI2||l°° - IMMI^oIIlp' = IMMMI^v < CIMMMI2^- Here 1/p + 1/p' = 1 and the condition p > max(d/2, 2) ensures that 2p' < 2*. Thus we have \\V(x) + (w * \u0\2)(x)d) On the other hand, the Hartree equation can be rewritten as {uo,A (-A)2 + V2 - 2(-A) - C(V2 + 1) >(l-£)((-A)2 + F22)-C£, Vee(0,l). Putting differently, PoHl!2 > (1 - ^W^Wl^) + HVVIIV) - C£y\\2L2, \/e £ (0,1), £ D{A). Thus 7^(lMk2 + II^||l2)||^I|d(a0) < IMk2 + II^IU2- In particular, we have c™(Rd) = D(Aq). Next, using the bound \\V1(x) + (w*\u0\2)(x)v\\L2 0 68 CHAPTER 3. HARTREE THEORY we conclude, by the Kato-Rellich theorem that A := —A + V{x) + (w* K|2)(a:) - f-i. is a self-adjoint operator on L2(Rd) with domain D(A) = D(A0). The Hartree equation can be rewritten as {uo,A R be a measurable function. Consider the Schrddinger operator A = -A + V(x) on L2(Rd) with D(A) = H2(Rd) nD(V). Prove that A is a self-adjoint operator. 3.6 Positivity of minimizers -\ Theorem (Positivity of Hartree minimizers). Consider the Hartree functional £n(u) := J (\Vu(x)\2 + V(x)\u(x)\2^dx + - ff \u(x)\2\u(y)\2w(x - y)dxdy JRd ^ ' 2 J JRdxRd with V+ G L™c(Rd), V-,w G Lp(Rd) + L°°(Rd), p > max(d/2,1). Assume that the minimization problem 4(A) := inf {s£(u) : u G H\Rd), \\u\\2L2{Wid) = \] 3.6. POSITIVITY OF MINIMIZERS 69 has a minimizer uq G H2(M.d) and it satisfies the Hartree equation ( - A + V{x) + (w * \u0\2)(x) - fiju0(x) = 0, for a.e. x G Rd. Then we have • Positivity of minimizer: There exists a constant z G C, \z\ = 1 such that zuq(x) = \uq{x) \ > 0, for a.e. x G Rd. Moreover, \uo\ is also a Hartree minimizer. • Positivity of mean-field operator: We have -A + V(x) + (w* \u0\2)(x) - [i > 0. Moreover, this operator has the ground state energy 0, and \u0\ > 0 is its unique ground state up to a phase factor (i.e. all ground state are given by z'\u0\ with z' G C, \z'\ = I). ) We start with recalling a very useful bound. Theorem (Diamagnetic inequality). For any u G i1/1(lRd), we have \u\ G i1/1(lRd) and \V\u\(x)\<\Vu(x)\, fora.e.x£Rd. This is equivalent to the convexity of gradient: for real-valued functions f,g G IVa//2 + 92{x)\2 < |V/(x)|2 + |V#(x)|2, for a.e. x G Rd. In the latter bound, if we have the equality \^Vf2+92(x)\2 = IW(x)|2 + |V#(x)|2, for a.e. x G Rd. and f{x) > 0 for a.e. x G Rd or g(x) > 0 for a.e. x G Rd, then f{x) = q/(x) for a constant c independent of x G Rd. Remarks: 70 CHAPTER 3. HARTREE THEORY • The the convexity of gradient holds true also for complex valued functions as VyVW+WW < |V|/|(x)|2 + \V\g\(x)\2 < \Vf(x)\2 + \Vg(x)\2. • A more general form of the diamagnetic inequality: For any given vector field A G L2oc(IR3,1r3) we have the pointwise estimate |VH(x)| < |(V + iA(x))u(x)\, for a.e. x G Rd. This explains the name "diamagnetic inequality". Proof. Step 1. Consider u = f + ig with real-valued functions /, g G H1(Md). Then we have the pointwise formula 'o Xu(x) = 0, V\u\(x) = I /(x)v/(x) + g(x)Vg(x) \u\x) if u{x) 0. In fact, for any e > 0 we define G£ = VM2 + e2 -e = v7/2 + g2 + e2 - e. Note that f2(x) + g\x) 0 < Ge(x) = ^- < y/p(x) + g\x) G L2(Rd) y/f2(x)+g*(x) + e2 + e ~VJKJ and f(x)Vf(x) + g(x)Vg(x) VGJx) = ^/f2(x)+g2(x) + e2 By the Cauchy-Schwarz inequality, we have the pointwise estimate ' v 71 - v//2(a;)+^2(a.)+e2 -vi jwi v ^ Since {Ge} is bounded in H1(Md) and Ge(x) —> \u(x) \ pointwise as e —> 0, we obtain G£ —^ |-u| 3.6. POSITIVITY OF MINIMIZERS 71 weakly in i/1(]R■ D := 0 if u(x) = 0, MI strongly in L2 (by Dominated convergence), we find that V|-u| = -D. Step 2. By the Cauchy-Schwarz inequality we have, when u(x) = fix) + ig(x) ^ 0, \f(x)Vf(x)+g(x)Vg(x)\ W\u\(x)\ = \u\x) = V\Vf(x)\2 + \Vg(x)\2 = \V Step 3. In the above Cauchy-Schwarz inequality, the equality |Vv7/2 + 92(x)\ = y/\Vf(x)\2 + \Vg(x)\2, for a.e. x G IRd. occurs if and only if f(x)Vg(x) - g(x)Vf(x) = 0, for a.e. x G Md. Now assume that g(x) > 0 for a.e. x G Rd (the case /(x) > 0 is similar). Then the above equality implies that ^ff{x)\ f(xWq(x) — q(xWf(x) „ , v (^-tt )= o/ x =0, for a.e. x G M. . \g(x)J g\x) Thus //g = c a constant. e.d. (Rd). J Now we apply the diamagnetic inequality to Hartree theory. Proof of the positivity of Hartree minimizers. Step 1. By the diamagnetic inequality we have £&{u) -£H(M) = ||Vií|||2 - ||V|ií||||2 > 0. 72 CHAPTER 3. HARTREE THEORY Therefore, if u$ is a minimizer for e^(A), then \u$\ is also a minimizer. Step 2. Now assume that uq > 0 is a minimizer for e^(A). We prove that uq(x) > 0 for all x G Rd. We write the Hartree equation as ( - A + W{x))uq[x) = 0 with W(x) := V(x) + (w* \u0\2)(x) - fjL(X) The special case V+ G L°°(Rd). Then because w * |ií0|2 G L°°(Rd), we have W+ G L°°(Rd). Thus we can take a large number m > 0 and rewrite the Hartree equation as (-A + m2)u0(x) = (m2 - W)u0(x) > 0, Vx G Rd. Since the operator (—A + m2)-1 is positivity improving, this implies that uq(x) > 0 for all x G Rd. Exercise (Positivity improving property). Let m > 0. Prove that the Yukawa potential satisfies K(x) := / -.---e27Tik-xdk = f°° /A exp ( - ^ - m2t)dt > 0. \2TTk\2+m2 j0 (4TTt)d/2 ťV At Deduce that ifO < g G L2(Rd) and g ^ 0, then ((—A + m2)~1g)(x) > 0 for a.e. x G The general case V+ G L™c(Rd). We have W+ G L^c(Md). Thus for any R > 0 there exists m > 0 large such that (-A + m2)u0(x) = (m2 - W{x))u0(x) > 0, V|x| < R. The strict positivity u(x) > 0 on |x| < R then follows from the following general result (see [Lieb-Loss, Analysis, Theorem 9.10] for a proof, and even a more general version). Theorem (Harnack's inequality). Let m > 0 and 0 < / G H2(Rd). Assume that (-A + m2)f(x) > 0, for a.e. \x\ < R. 3.6. POSITIVITY OF MINIMIZERS 73 Then f(x)>c0 / f(y)dy, for a.e. \x\ < r < R. J\y\ 0 is independent of f. Since uq > 0 and u$ ^ 0, we can choose R > 0 large enough such that u0(y)dy > 0. '\y\ 0 for |x| < R we find that ^oO^) co / uo{y)dy > 0, for a.e. |x| < f?/2. ■%l oo, we obtain uq(x) > 0 for a.e. x G lRd. Step 3. Assume that uq > 0 is a strictly positive solution to the Schrodinger equation (-A + W{x))uq{x) = 0, x G Rd. Then 0 is the ground state energy of —A + W(x) and uq is a ground state. This follows from the following general fact. Theorem (Perron-Frobenius Principle). Let 0 < / G H2(Rd), W G Ljoc(Rd) such that -Af(x) + W(x)f(x) = 0, for a.e. x G Rd. Then —A + W > 0, namely [ f|V^|2 + w^M2) > 0, V^GCCC Proof. Since / > 0, we can define g = ip/f. Then substituting ip = fg we find that f |Vyf = f \v(fg)\2 = f \f(Vg)+g(Vf)\2 = f [l/l2|v^|2+|^|2|v/|2+2K7(v^^(v/) Moreover, by integration by part H'm2 = - [fdmm2) = - [f(din\g\2 - [mmm2) 74 CHAPTER 3. HARTREE THEORY and hence by summing over j = 1, 2,d I g\2Nf\2 = Jf(-Af)\g\2-2 j fVm(g(Vgj) J |V^2 = / I/IW + J f(-Af)\g\2. In summary Therefore > 0. =o q.e.d. Since C£°(M.d) is the core domain of —A + W, the above quadratic form estimate ensures that —A + W > 0 as an operator. Step 4. Now let us conclude. Assume that uq is a Hartree minimizer. By Step 1, \uq\ > 0 is also a Hartree minimizer. By Step 2, \u0\ > 0 pointwise. By step 3, both u0 and \u0\ are ground states for the Schrodinger operator —A + Wix). Let us prove that |'Uo(a:)| = zu0(x) for a constant z G C independent of x G Md. In fact, we can write u$ = f + ig with real-valued functions /, g. Then /, g are also ground states for — A+W(x). By the diamagnetic inequality, |/| is also ground states for — A + W(x). Since |/| > 0, arguing as in Step 2 we conclude that |/| > 0. Next, let h G {f,g}- Since h and |/| are ground states for —A + W(x), the function is also a ground state for —A + W(x). By the diamagnetic inequality again, |$| is also a ground state for —A + W(x), and moreover we have the equality Since |/| > 0, the equality case in the diamagnetic inequality tell us that h(x) = c^|/(x)| for $:=fc + i|/| namely V^/h2(x) + f2(x) \ = ^J\Vh(x)\2 + |V|/|(x)|2, for a.e. x G Rd. 3.7. UNIQUENESS OF MINIMIZERS 75 a constant independent of x. Thus uo(x) = fix) + ig(x) = cf\f(x)\ + icg\f(x)\ = (cf + icg)\f(x)\ with Cf + icg independent of x £ Rd. This implies that zu0(x) = \u0(x) \ for a constant z £ C independent of x £ Rd. q.e.d. 3.7 Uniqueness of minimizers In general, uniqueness is a hard question, and the answer depends a lot on the potentials. In this section we will focus on a simple case where the interaction potential is of positive-type, making the Hartree functional convex. / Definition (Positive-type potential). A potential w : lRd —» R is of positive-type if w(x — y) is the kernel of a positive operator on L2(M.d), namely {f,w*f) = JJ f{?)f{y)w{x - y)Axdy > 0. This property is equivalent to w > 0 because {f,w*f)L2= f(k)w*f(k)6k= \f(k)\2w(k)dk. Theorem (Uniqueness of Hartree minimizers). Consider the Hartree functional £k(u) := / ('lV-u(x)l2 + V(x)|it(x)|2V)dx H— ff \u(x)\2\u(y)\2w(x — y)dxdy with V+ £ L™c(Rd), V-,w £ Lp(Rd) + L°°(lRd) with p > max(d/2,1). Assume further w(k) > 0, for a.e. k £ Rd. Then the followings hold true. • Convexity of the functional: 0 < p i—> £^(^/p) is convex, namely for p\ > 0, 76 CHAPTER 3. HARTREE THEORY p2>0 such that ^/p~{, ^/p^ G H1^) and for t G [0,1] t^(vpl) + (1 - t)£l(^) > El (^JtPl + (l-t)p2). Uniqueness of minimizers: For any A > 0 the minimization problem 4(A) :=mf{^ u) : u G H l nn>d\ u L2 = A has at most one minimizer u0 > 0. This minimizer is unique up to a phase factor (i.e. all minimizers must be given by zuq with z G C, \z\ = 1). Proof. Step 1. Let px > 0, p2 > 0 such that ^/p~[, ^fp^ G H1^). For any t G [0,1] we have 1-t p1(x)p1(y)w(x - y)dxdy + P2(x)p2(y)w(x - y)dxdy tpi(x) + (1 - t)p2(x) tpi(y) + (1 - t)p2{y) w(x - y)dxdy = t(l-t) J J (pi{x) - p2{x)^j (pi(y) - p2{y)>)w(x -y)dxdy > 0. In the last estimate we used the fact that w is of positive-type. Combining with the convexity of the gradient term * J |VvpT|2 + (i-t) J\v^/p~2\2> J |VvV + (i-*)p2|2 we find that for all t G [0,1], t£V(y/pT) + (1 - t)£V{Jfr) > El {^tPl + (l-t)p2). Step 2. Assume that 4(A) has minimizers uq,vq. By a previous theorem, we know that zuq{x) = \uq{x)\ > 0 and z'vq(x) = \v${x)\ > 0 with phase factors z,z' G Z, \z\ = \z'\ = 1. Moreover, \uq\ and \vq\ are also minimizers for 4(A), thanks to the diamagnetic inequality. It remains to prove that \uo\ = \vo\. By the above convexity of the Hartree functional, we have 4(A) = ^(m) + l^(\vo\) > £l(\j\\uo? + \\vo?) > 4(A)- 3.7. UNIQUENESS OF MINIMIZERS 77 Here the last estimate follows from the variational principle and the constraint Thus we must have ^(Kl) + \f%(\M) = el U\\M2 + ^M2 which in particular implies that for the gradient terms 1 iVliiol 1 |VMI2 = 2 J 2 By the diamagnetic inequality this means that 1 1 2 W-M2 + 2M2 1 1, 2,vM(x)|2 + -|v|^o|(^)|2 = vW-KWI2 + -h(x)|2 1 1 for a.e. x E Rd and because \uq\ > 0, \vq\ > 0 we must have \uq\ = c\vq\ for a phase factor c. Since uo\2 = / M2 = A the phase factor is c = 1. Thus we conclude that \uo\ = \vo\. This completes the proof, q.e.d. Note that in the above theorem, we did not discuss the existence of minimizers. In the trapping case Vix) —> +00 as |x| —> 00, the existence of minimizers is guaranteed. However, in the vanishing case Vix) —> 0 as |x| —> 00, it may happen that the minimizers do not exist if the mass is large enough. Using the convexity property, we can also prove the existence of a critical mass Ac, where a minimmizer exists if and only if A < Ac. Theorem (Convexity and the critical mass). Consider the Hartree functional £h(u) := / r|V-u(x)|2 + l/(x)|-u(x)|2N)dr H— // \u(x)\2\u(y)\2w(x — y)dxdy ./rod V / 2 / /radvIBd and the Hartree energy e£(A) := inf |^(ii) -.uEH^ = A } 78 CHAPTER 3. HARTREE THEORY with V,w G Lp(Rd) + Lg(Rd) with oo > p, q > max(d/2,1) and w(k) > 0. Then the fallowings hold true. • The mapping X —> e^(A) is convex and decreasing. Consequently, there exists a critical value 0 < Ac < oo such that e£(A) > el(Xc) = e£(A'), VA < Ac < A'. Here we used the convention e^(Q) = 0 and e^(oo) = oo. • The minimization problem e^(A) has a minimizer if and only if A < Ac. Proof. Exercise! q.e.d. Let us give an example where the previous abstract theorems apply. Theorem (Hartree minimizers for bosonic atoms). Let Z > 0 and consider the Hartree functional for atoms £h(u):= [ (\Vu(x)\2-^\u(x)\2)Ax + - if H^lX?/) 7r3 V \x\ j 2JJM3X-M3 \x — y\ and the Hartree energy E(Z, A) = inf {Sniu) \ u G H\R3), [ \u\2 = x\. Then there exists a critical mass Xc = \C(Z) G [Z, 2Z) such that E(Z, A) has a minimizer if and only if X < Xc. Moreover, the minimizer is strictlly positive, unique up to a phase, and radially symmetric. Proof. The Coulomb potentials satisfy all relevant conditions, in particular |^P(A;) = 4tt|A;|-2 > 0, k G R3. The existence of the critical mass Ac thus follows. The lower bound XC(Z) > Z has been proved before. The upper bound Ac < 2Z is an exercise (c.f. Lieb's non-existence theorem for many-body Schrodinger theory). From the above discussion, we know that when exists, the minimizer u$ is strictly positive and unique up to a phase. Morever, since the external 3.8. HARTREE THEORY WITH DIRAC-DELTA INTERACTION 79 potential Vix) = —Z/\x\ is radially symmetric, the Hartree functional u \—> £^(u) is rotation-invariant. Therefore, the unique minimizer u0 > 0 must be radially symmetric. q.e.d. Remark: For bosonic atoms the critical mass is \C(Z) = \.2\Z. The linearity on Z can be seen easily by scaling (how?). The value 1.21 is numerical. As we will see the behavior \C(Z) ~ 1.21 Z also holds for the many-body Schrodinger theory in the limit Z —> oo. Exercise. Denote ca := it a^2T(ya/2) with the Gamma function oo z-l„-t. T(z) = / ť-^dt. 0 (Note that T(n) = in — 1)\ for n 6 N.) Prove that for all 0 < a < d we have ^L. = ^Z£^, VA: G Rd. \rjQ \ot a—a. Hint: You can write e-^Xa/2-ldX 0 \x\" and use the Fourier transform of the Gaussian. 3.8 Hartree theory with Dirac-delta interaction So far we have study the Hartree theory with regular interaction potentials. The method represented in this chapter can be adapted to treat Dirac-delta potentials, which model short-range interactions appear often in physical set up. In this case, the Hartree theory is often called the Gross-Pitaaevskii theory or nonlinear Schrodinger theory. Exercise. Consider the Hartree functional with Dirac-delta interaction ^(u):= / (\Vu(x)\2 + V(x)\u(x)\2 + -\u(x)\4)dx with a constant a > 0 and a function V : Rd —> R satisfying V+ G Lfoc(lRd), V- G Lp(Rd) + Lq(Rd), oo > p, q > max(d/2,1). 80 CHAPTER 3. HARTREE THEORY For every A > 0 define 4(A) := inf {£l(u) : u G H\Rd), \\u\\2L2{WLd) = x}. 1. Prove that ifVix) —> +oo as \x\ —> oo, then 4(A) has a minimizer for all A > 0. 2. Prove that ifVE Lp(Rd) + Lq(Rd) and the strict binding inequality holds e£(A) < 4(A'), V0 < A' < A, then 4(A) has a minimizer. 3. Prove that i/e^(A) has a minimizer, then it has a unique non-negative minimizer. (Hint: 0 < p i—> £^(-v/p) is strictly convex^. I J Chapter 4 Validity of Hartree approximation In this chapter we will derive rigorously the Hartree theory as an effective description for many-body quantum systems. We start from many-body quantum mechanics. Consider a system of N identical bosons in IRd, described by the Hamilttonian n HN = Y(-AXi + V(xi)) + A ^ w(Xl - xj) i=l KkjXN acting on L2(M.d)®sN. As usual, V, w : lRd —> R and w is even. The parameter A > 0 is used to adjust the strength of the interaction. In this chapter, we will focus on the mean-field regime A = 1 N - 1 In this case, the Hartree functional obtained by taking expectation against the product state u®n -g independent of N: — {u®N,HNu®N)= f (\Vu(X)\2+V(X)\u(X)\2)+- [[ \u{X)\2\u{y)\2w{X-y)dxdy=:S^{u). N jRd \ J 2 J J We consider the ground state energy of Hn EN := inf HN^) \L2md-}®aN = 1 and the Hartree energy eH := inf En(u) IMIl^i-Rd)-! 81 82 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION We will prove that under appropriate conditions on V,w, the Hartree theory describes correctly the leading order behavior of the ground state energy and the ground states of HN when N —> oo. 4.1 Reduced density matrices Definition. For any wave function ^N 6 L2(M.d)®sN and any 0 < k < N, we introduce the /c-body reduced density matrix 7^-' this is an operator on L2(M.d)®ak with kernel (k) / / N\ f %>(z1,...,zk;z[,...,zk) = —-— / ^N(z1, ...,zk,xk+1,...,xN)x (N — k)\ JRd(N-k k) X \&at(4, z'k, xk+1,xN)dxk+1...dxN. Equivalently, we can interpret 7^ as the partial trace over all but the first k variables Note that 7^ is a non-negative, trace class operator on L2(M.d)®ak and (N-k)l' For example, the one-body density matrix 7^ is the operator on the one-body space L2(Rd) with kernel %' (x,y) := N / ^N(x,x2,...,xN)^N(y,x2,...,xN)dx2...dxN. XN-l) Its diagonal part is called the one-body density p\siN(x) := N / \^n(x, x2,XN)\2dx2...dxN. (N-l) The function p^N is the probability distribution of the particle density, namely JQ p^N can 4.1. REDUCED DENSITY MATRICES 83 be interpreted as the expected number of particles in £1 C Rd; in particular / P^N = Tr7 (i) = N. JWLd N Exercise. Consider a wave function ^N £ L2(M.d)®sN and a one-body operator h on L2(Rd). Prove that N i=i Moreover, prove that for any multiplication operator Vix) on L2(R.d) regular enough (e.g. V £ C™(Rd)) we have Tr(y7£)= / V(x)p*N(x)dx. Exercise. Let 7 be a non-negative trace class operator on L2(M.d) with the spectral decomposition 7 = ^2n>1 \n\un){un\. We define its density as unlx n>l Prove that if ~fn —> 70 strongly in trace class, then pJn —> p10 strongly in L1(Rd) Remark: In physics litterature the density of an operator is often written as p7(x) = 7(x, x). Mathematically the kernel of an operator on L2(lRd) is often defined for a.e. (x, y) £~Rd x lRd, making the discussion on the "diagonal part" 7(x, x) a bit formal (as the set {(x,x) £ Md} has 0 measure in lRd x However, using the spectral decomposition 7 = ^2n>1 Ara|-ura)(-ura| we can properly define the kernel 7(x,y) = \ \nunix)un{y) n>l which makes sense for a.e. x, y £ M.d, and hence the formula p7(x) = 7(x, x) becomes correct. An equivalent way to define the density p7 without using the spectral decomposition is to use the formula Tt(Vj) = / l/(x)p7(x)dx 84 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION for all regular multiplication operators V(x) on L2(IRd) The energy expectation of n HN = ^2(-AXi + V{xi)) + A ^ w(x% - Xj) i=l KkjXN can be rewritten conveniently using the one- and two-body density matrices HN*N) = Tr((-A + V)^) + £ Tr^g)- The complexity of the iV-body problem lies on the fact that it is very difficult to characterize the set of all two-body density matrices for N large. The so-called the N-representability problem is (quantum) NP hard, see e.g. a paper of Liu, Christandl, and Verstraete (PRL 2007). On the other hand, the set of one-body density matrices is well-understood. Exercise. Let 7 > 0 be a non-negative, trace class operator on L2(Rd) with Tr7 = N. Then there exists a wave function ^N G L2(Rd)1 Xn\un){un\ Vou can choose n>l The key idea of the mean-field approximation is to replace the complicated two-body density matrix 7p^ by the tensor product of the one-body density matrix 7*^ ~ 7*^ <® 7*^ • For the ground state energy of the Hamiltonian Hn, we will even go down to the level of one-body density p^N and try to prove that ^N,HNUn^£u(J^), 4.2. HOFFMANN-OSTENHOF INEQUALITY which eventually leads to the validity of the Hartree theory 85 4.2 Hoffmann—Ostenhof inequality The approximation N ' ' Vy N 7' is nontrivial even for non-interacting systems. For the external potential, we have the exact identity n (*N,y2v(Xl)*N) =Tr(V^N) = / V(x)pyN(x)dx. N i=i ' J^d However, for the kinetic operator, in general we have }Z ~^n) = Tr(-A7« ) + / | V^|2. i=l Nevertheless, we still have the following sharp lower bound, which will be very useful to justify the Hartree approximation. -\ Lemma (Hoffmann-Ostenhof inequality). For every wave function f« G L2(M.d)®sN we have n „ *N,J2(-AXi)*N)>(y/p^;,-Ay/p^)= |vy7^-;|2. i=i ' d Proof. Step 1. Since the one-body density matrix 7^ is a non-negative trace class operator, we have the spectral decomposition n>l with an orthogonal family {fn}n>i C L2(IRd) (the functions fn are not necessarily normalized). Then we have the one-body density [x)\2. n>l 86 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION Thus the Hoffmann-Ostenhof inequality is equivalent to Y [ \v.Ux)\2dx> f v/£ !/„(*)! n>l d V n>l dx. Step 2. We prove the latter bound for finite sums m » Y / |V/„(x)|2dx> \ n=l dx, Vm = 1, 2, ... We can prove that by induction in m. The case m = 1 follows from the diamagnetic inequality > |V|/i|(x)|. For m = 2, by the diamagnetic inequality we have |VA(x)|2dx+ / |v/2(x)|2dx> / VvUW+LMxTl dx. For m = 3, using the diamagnetic inequality twice we have |VA|2+/ |v/2|2+/ |v/3|2>/ vVW+W2+f |v/3 JKd JKd JKd JKd > vVl/il2 +1/2|2 +1/3| The same applies to other values of m. Step 3. In principle passing from finite sums to infinite sum should be easy thanks to standard density arguments. Let us explain it here. Of course it suffices to consider the case when the left side is finite. For any m > 1, denoting \ 1^" \ 71=1 'xW. Then {gm}m>i is an increasing sequence and \ n=l 0 < gm{x) < _ \ n=l Therefore, by Lebesgue Motonone Convergence Theorem, gm —> ^p^N strongly in L2(lRd) as 4.2. HOFFMANN-OSTENHOF INEQUALITY 87 m —> oo. On the other hand, we have proved in Step 2 that » m „ oo „ / |V#m|2 0 on L2(lRd) satisfying {u,hu)> {\u\,h\u\), V-u G L2(lRd) then we have the Hoffmann-Ostenhof inequality N i=i The condition {u, hu) > {\u\, h\u\) is equivalent to each of the following statements. 1. The resolvent (h + C)_1 is positivity preserving, namely it maps positive functions to positive functions 'f{x) > 0 for a.e. x G => ({(h + C)'1 f)(x) > 0 for a.e. x G 2. The operator e~th is positivity preserving for all t > 0. Exercise. Let h > 0 be a self-adjoint operator on L2(R.d) such that (/(a:) > 0 for a.e. i6RJ^ ((e-thf)(x) > 0 for a.e. iGBJ, W > 0. Prove that for any function u belongs to the quadratic form domain of h we have {u, hu) > {\u\, h\u\). 88 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION Since the heat kernel etA(x;y) is positive, the operator eiA is positivity preserving for all t > 0. Thus the above exercise gives an alternative proof of the diamagnetic inequality 4.3 Onsager's lemma Now we consider the approximation from the angle of the interaction terms. A simple but very useful observation is Lemma (Onsager's lemma). If 0 < w G L1(Md), then for all 0 < g G L1(lRd) we have the pointwise estimate N ^ w{xl - xj) > ^{g * w)(xj) ~ 2 / / g(x)g(y)w(x - y)dxdy - —w(0). l^ G L2(M.d)®sN we have Vn, Y w(xl - Xj)^N^ > \ I I P*N(x)P*N(y)w(x - y)dxdy - yw(0). l(0) is of order N, which is much smaller than the main term. Proof. Step 1. Since w > 0, the potential w is of positive-type. Therefore, f(x)f(y)w(x - y)dxdy = (/, w * /) > 0 for any "reasonable function" /. By choosing N 1=1 with 5q the Dirac-delta function and using the identity 5q * p = p we obtain the pointwise 4.3. ONSAGER'S LEMMA 89 bound N / J w(xi-xj)>2_^(g*w)(xj) 1(/c) > 0, we can complete the square AT ^ ^ ^2-Kik-Xj N > 2^g{k)Ye 3 = 1 l?(*)ls and find that -2jm\Y dk > "ü I w{k)g{k)Ye2mk'X3dk ~ ~ w(k)\g(k)\2dk N = ^Y w*~9(k)e2wk'Xjdk - - / w{k)\g{k)\2dk N 1 = YW * 9(xj) - 2 J w{k)\g{k)\2dk. Step 2. Now we apply the above pointwise estimate with g = p^N, and then take the expectation against ty^. We obtain AT Y W(Xi - xj)^n) > (^N, Y(W * P^n)(xj)^N l 2 and {x}^=1 C Rd we have / , N2 f N , x 2^ w[x, -Xj)>— w- —w(0). l p, q > max(d/2,1). Let En be the ground state energy of N 1 i=l loo En/N is exactly en- Step 2. Now we consider the "easy case" 0 < w G L1(lRd). For any wave function ^N £ L2(M.d)®3N, by the Hoffmann-Ostenhof inequality we have N ^N,^2{-AXi + V{xi))^N^ > f |V-/P^|2+ / Vp*N i=i J •> and by Onsager's lemma we have 1 11/*/* -/V — (^n, w(xl ~ ^O^Jv) > _ 1 2 // P*N(x)P*N(y)w(x ~ V)dxdy - —w(0) N _ . l N£u (-y ^) - C > Neu - C. In the last estimate we have used the variational principle for en- In conclusion we have Neu > EN > Neu - C which implies the desired convergence En/N —> en- Step 3. Now we consider the case w G L1(Md). Since w has no sign, we will use Onsager's lemma for its positive and negative parts separately. The proof below is due to M. Lewin, 92 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION using ideas of Levy-Leblond and Lieb-Yau. We decompose W = Wi — W2, Wl = (w)+ > 0, W2 = (w)- > 0. It is more convenient to consider E2N. Take a wave function ty2N £ L2(]Rd)lX>s2Ar. Then using the bosonic symmetry we can rewrite the expectation (XS?2N, H2N^2N) as follows. For the one-body terms, we have 27V ^>2N, J^(-AX! + V(xl))^>2N) = 2N(*2N, (-AX1 + V(x1))^>2N) i=i n = 2^2N, ^2(~AXi + V(xi))*2N i=i For the interaction terms involving wi, we write ^rr(^2AT, Wl(Xi ~ xi)^™^ = N(^2N,w1(x1 - x2)ty2N^ "Y^2Ar, ^ wi(xi-xj)^ 2N _ l Neu - C. Indeed, for any wave function £ L2(M>d)®sN, by the Hoffmann-Ostenhof inequality and Onsager's lemma (twice) we have i=l l f iVVPi^l2 + J VP^N + j^—^ N i rr N - pq>N{x)pq>N(y)w1(x - y)dxdy - —w^O) 1 ■s—^ Iff N Y(a * w2){yj) - 2 J J g(x)g(y)w2{x - y)dxdy - —w2{o) N - 1 1 w 3) for any function 0 < (/ G L (R ). By choosing iV- 1 9 = N we have and hence N 1 N 1 N — Y(a * w2)hjj) - Y^" * W2^y^ =0 $N,HN$N) > / |vy7i 2(N -1) pN{x)pq>N(y)w1(x - y)dxdy (JV-1) ff N ( --Ž/V2- J J P®N ^P^N (y^2 (X ~ y)dxdy - 2(N -1) V 1 ^ + W2 ^ > J\^Vp^\2 + J VP^n + 1^ J J p^Ni.x)p^NÍ.y)wÁx - y)Axdy 94 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION Iff ........ N 1\ ( pq,N {x)pq>N (y)w2 (x - y)Axdy - _ (wi (0) + w2 (0) 2N jj ^nk/^nky/zk y/ » 2(N-1) Since this holds for any wave function §N G L2(M.d)®3N, we conclude that for any given yi,...,yNeRd, HN > Neu - C. Consequently for any wave function £ L2(Rd)®s2N we have {^2N, H2N^2N) = 2{^2N, HNm2N) > 2Neu - C. Therefore, E2N > 2Neu - C, and hence . E2N C w,fN1 Since N i—> E^/N is increasing, we conclude that eH>f >eH-§ W>1. This concludes the proof of lini7v->oo E^/N = en when w G L l(T0>d\ Step 4. Now we consider the general case w G Lp(Rd) + Lq(Rd) with oo > p, q > max(d/2,1). Then we can take we G C£°(M.d) such that we —> w in Lp + Lq as e —> 0+. More precisely, we write w = f + g, f G Lp(Rd), g G Lq(M.d) and choose we = fe + ge, f£, g£ G C™(Rd), \\fe - f\\LP + \\ge - g\\Lq < e. We take a wave function ^ N G L2(RdN) such that(vpat, HN^N) < CN. Then using 1 W i=i 4.4. CONVERGENCE TO HARTREE ENERGY (why?) we have the a-priori estimate N *n,Y(-a^n) max(d/2,1) implies that 2p' G (2, 2*). Similarly, {Vn,\(gs ~ g)(xi ~ x2)\^N) < C\\g£ - gUu^i^n, (1 - AX1)^N). Using the choice of f£, g£ and the bosonic symmetry we deduce that 1 N Y {*Nd™-™e)(xi-Xj)*N)>-Ce^{*N,(l-AxA*N)>-CeN. l Ne^£ - C£ i=l l- J £ J J > Thus in summary, we obtain the lower bound {*N, HN^N) > Ne^£ -C£- CeN for any wave function satisfying {^^,H^^) < CN. This implies that lim ^ > eH £ - Ce, Me > 0. 96 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION The conclusion follows from the fact that en,e —> en as e —> 0 (exercise). q.e.d. Exercise. Assume that V+ G L™c(Rd), w0, V- G Lp(Rd) + Lq(Rd), oo > p, q > max(d/2,1). Let w£ —> wo in TP + Lq as e —> 0+ and define the Hartree energy eH,£:= inf { / (\Vu\2 + V\u\2) + J / / |ii(x)|2|ii(y)|V(x - y)dxdy\. HullL2(Rd)-l *• JE,d I J jRdxRd J Prove that en £ ~> eH o as e —» 0+. 4.5 Convergence to Hartree minimizer Now we turn to the convergence of ground states. Heuristically, if is a ground state of HN, or more generally an approximate ground state, i.e. (VN, HN^N) = EN + o(N) then we expect that where uq is a Hartree minimizer. Here the approximation ~ u®N means that most of the particles in the iV-body state occupy a common one-body state u$. This phenomenon is called the Bose-Einstein condensation (BEC). Note that the approximation ~ u®N has to be understood in an appropriate sense. In fact, and u®N are not close in the usual norm of the Hilbert space L2(M.d)®3N (except the non-interacting case). The proper meaning of the Bose-Einstein condensation can be formulated in terms of reduced density matrices. Definition. Consider the quantum states {^n}n>i, where is a wave function in L2(M.d)®sN. We say that there is the complete Bose-Einstein condensation if there 4.5. CONVERGENCE TO HARTREE MINIMIZER 97 exists a one-body state uq G L2(Rd) such that lim N 1. Remarks: J • If ^ n u, , then 7- = N\uq)(uq\, and hence (^0,7^0) N 1. • For any wave function £ L2(Rd)lg,sAr, {uq,^' uq) is interpreted as the expectation of the particle number in the condensate state u0. In general, we always have • By the variational principle, the complete BEC is equivalent to the fact that the largest eigenvalue of 7^ is N + o(N). Further equivalent statements of the BEC are in the following exercise. Exercise. Consider the quantum states {^n}n>i, where is a wave function in L2(M.d)®sN. Let uq G L2(M.d). Prove that the following statements are equivalent. 1. limAr^ooiV-1 («0,7*^0) = 1- 2. iV_17^ —> \uo){uo\ strongly in the operator norm. 3. iV_17^ —> \uo){uo\ strongly in the trace class norm. Hint: A = iV_17^ — |-uq)('uo| has trace 0, and exactly one negative eigenvalue (except (u0,1^nu0)N + o(N). J In principle, the BEC is not equivalent to convergence of ground states. In fact, proving the BEC is often more difficult than proving the convergence of ground states. 98 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION Theorem (Convergence to Hartree minimizer: "easy case"). Assume that • 0 < w G L1(Rd); • V G 1^(1^) and V{x) —> +oo as \x\ —> oo. Let ^at G L2(Rd)®sN be an approximate ground state of N 1 HN = ^(-AXi + V{xi)) + _ - Xj). i=l l N _ 1 2 // P*n(x)P9N(y)w(x ~ y)dxdy - ytu(O) ^^11 P^n(x)P^n(y)w(x ~ y)dxdy - C. N . l For the kinetic term, we do not use the Hoffmann-Ostenhof inequality. Thus we get {vN,HNyN) >Tr((-A + F)7i1i) + 2^ //P*iv(^)P*iv(yM^-y)^dy-C. Here we keep the one-body density matrix because it is the relevant object for the BEC. Step 2. The new idea now is to linearize the non-linear term. Since w is of positive-type, we can use f(x)f(y)w(x - y)dxdy > 0 with / = p^N(x) — N\uq\2 with uq the Hartree minimizer. This gives IS P^n(x)p^n(y)w(x - y^dy - ff P^n(x)\uo(y)\2/w(x - y)dxdy - y ff \u0(x)\2\u0(y)\2w(x - y)dxdy. 4.5. CONVERGENCE TO HARTREE MINIMIZER 99 Recall that under our conditions on w, V, the existence and uniqueness of the Hartree mini-mizer u0 have been proved in the previous chapter. Moreover, we have the Hartree equation hmfUQ = 0, /imf := —A + V + (w * \u0\2) — [i with the chemical potential „ = / (|V«,(I)P + v(*)M*)| > + // K(*)|>K(»)IM* - vWy /rod V / ././rodvPrf Thus eH + - // K0*0| K(y)| wix -y)dxdy. 2^ / / P*N(x)P9N(y)w(x ~y)dxdy > II p^N(x)\u0(y)\2w(x - y)dxdy - y // |ii0(a:)lVo(y)|2^(^ - y)dxdy = y P*iv(m2 * + N(eu - fi) = Neu + Tr (^\u0\2 * w - fi)j^ Combining the latter bound with the previous bound on ( H^^n ) we deduce that HNVN} > Tr ((-A + Fhfi) + ^ //P*n(x)P*n(yMx ~ y)dxdy - C >Ne^ + Ti{hma(Z)-C- Thanks to the uniform upper bound y&n, HN^Nj < Ne^, we conclude that Tr (^7£) < C. Step 3. From the Hartree theory, we know that the one-body Schrodinger operator /imf := —A + V + (w * \u0\2) — [i has the lowest eigenvalue 0 and u$ is its unique ground state (up to a phase factor). Moreover, since V(x) —> oo when |x| —> oo, /imf has compact resolvent. Thus it has eigenvalues 0 = Ai < A2 < A3 < 100 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION The strict inequality Ai < A2 is called the spectral gap. By the min-max principle, if we introduce the projection P=\u0){u0\, Q = l-P, then we have hmfP = Phm{ = 0 and hm{Q = QhmiQ > (A2 - Xi)Q. Next, let us decompose t£ = (p + QhSl (P + Q) = PiSIp + PiSlQ + QiSIp + Qi^NQ- Then by the above properties of hmfP, hm{Q and the cyclicity of the trace we have Combining with the previous bound Tr (h^aSl) < C-and the spectral gap A2 — Ai > 0 we conclude that C > Tr (g7g) = Tr ((1 - P)7g) =N- {u0,7^0). This completes the proof of the BEC. q.e.d. Theorem (Convergence to Hartree minimizer: general case). Assume that • V+ G ZC(Rd), w, V- G LP(Rd) + Lq(Rd), 00 > p,q> max(d/2,1). • The Hartree problem en = inf||u|| 2=i £n(u) has a unique minimizer u$ (up to a phase factor). Moreover, any minimizing sequence of en has a subsequence converging to uq (up to a phase) strongly in L2(M.d). Let G L2(M.d)®sN be an approximate ground state for N 1 i=1 l 00 as |x| —> 00, or V G Lp(M>d) + Lq(M>d) and we have the strict binding inequality 4(1) 0, we define the perturbed N-body Hamiltonian n HNiE = HN + e^PXs, P=\u0){u0\ i=i and call E^,e the ground state energy of H^,e- We prove that the complete BEC follows from the following claim: liminfliminf——-— > 1. e^0+ AT->oo eN Indeed, assume that the latter inequality holds true. Let ^at G L2(M.d)®sN be an approximate ground state for Hn- Then by the variational principle, we have e{uo, %Nu0) = {VN, HN:£tyN) - {VN, HNVN) > EN:£ — EN + o(N), and hence .. . c {uq,1{^nu0) . ENtS -EN + o(N) ^ 1 hm ml- > hm ml iim ml- > 1. TV-^oo N e-J>0+ 7V-J>oo eN 102 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION Step 2. Now we estimate En,£ ~ En- By the convergence to Hartree energy, we know that EN = Neu + o(N) where eH := inf j / (|Vn|2 + V\u\2) + - / / \u(x)\2\u(y)\2w(x - y)dxdy\. HUllL2(Rd)=l W]Rd 2 J JKdxRd J For our purpose, it is useful to introduce the Hartree energy for mixed states inf { Tr((-A + V)i) + \ [[ p7(x)p7(y)K;(x - y)dxdy}. on L (K ) I- 2 JJRdxRd J eH := 7>0 on. Tr 7=1 Here recall that p7(x) = 7(2;, x) is the density of 7 (defined properly by spectral decomposition). At first sight, it does not look very useful because en and en coincide! [ Exercise. Prove that e„ =5,. ffi„f Km can use the Hoffma„n,0.,te„hof .neauaHty. ) However, the advantage of &h is that its definition can be extended easily to the perturbed problem. For any e > 0, we define inf <^ Tr((-A + V + eP)j) + - p1(x)p1(y)w(x — y)dxdy >. on L2(Rd) y- 2 JJRdxRd J [r 7=1 This is the relevant limit of the perturbed N-body energy ENe- 6h,£ := 7>0 on Tr 7=1 Exercise. Prove that ,. ENe _ lim —— = eH,e. Hint: You can follow exactly the proof of the "Convergence to Hartree energy", without using the Hoffmann-Ostenhof inequality. Thus we have proved that for any e > 0, ,. En,£ — En ~ hm--- = eH,£ - eH. Step 3. Finally, we prove that eH,e - eH . iim - = 1. e^0+ e 4.5. CONVERGENCE TO HARTREE MINIMIZER 103 The upper bound en,e < en + e follows by choosing the trial state 7 = |wo)(^o| f°r eH,e- It remains to prove the lower bound. Let 7e be an approximate ground state for eH,e, namely 7e > 0 on L2(M.d), Tr(7e) = 1 and £h,£(7e) := Tr((-A + V + eP)7e) + \ [[ PlE(x)Pi.(y)w(x ~ y)dxdy < eH,e + o(e). The key point is to prove that Tr(Pj£) = (u0,j£u0) ->■ 1. If the latter convergence holds, then by the Hoffmann-Ostenhof inequality we find that £h,£(7e) > ^(-v/PtT) + £(uo, leUo) > eH + e + o(e) and the lower bound en,e > eH + e + o(e) follows. Thus it remains to show that {uo, 7e"Uo) ~~> 1- Convergence of density p7s. By the Hoffmann-Ostenhof inequality and the upper bound 6h,e 5: eh + we find that £h(VaP < ^(Ta^) + eTr(P7£) < £^£(%) < eH,£ + o(e) < eH + 0(e). Thus is a miminizing sequence for en when e —> 0+. Thanks to the assumption on the pre-compactness of minimizing sequences for en, up to a subsequence as e —> 0+ and up to a phase factor of uq, we have plE —> uq > 0 strongly in L2(M.d). Since plE is bounded in i1/1 (IRd) (as £n{y/p^) is bounded), by Sobolev's inequality we obtain r fTn>d\ ^/T-; ->■ ii0 strongly in Lr(Md) for all r G [2, 2*). Linearized equation. Using ^fp^~E —> uq in Lr(M>d) for all r G [2,2*) and the assumption w E Lp + Lg, we get \ II (PiAx) - \Mx)\2)(PiAy) - \My)\2)w(x - y)dxdy ->■ o which is equivalent to \ if PiAx)PiAy)w{x-y)dxdy= [[ \u0(x)\2plE(y)w(x -y)dxdy 104 chapter 4. validity of hartree approximation ~\ jj \uo(x)\2\uo(y)\2w(x - y)dxdy + o(l) = J(K|2 * w)plE + eH - p where „ = / (|v«,(l)P + v(*)M*)| > + // K(*)|>K(»)IM* - »)d»fc, JWLd ^ ' J JWLdxWLd is the chemical potential in the Hartree equation hm_{UQ = 0, /imf := — A + V + (w * \uo\2) — p > 0. Thus £-a,e(le) = eH + Tr(/imf7e) + e(ii0,7^o) + o(l). Consequently, Tr(/imf7e) ->■ 0. Weak-limit in Hilbert-Schmidt topology. Using the operator lower bound /Vf > ~\a - c we find that Tr((l - A)7e) = Tr((l - A)1/27e(l - A)1/2) < c. Thus (1 — A)1/27e(l — A)1/2 is bounded in trace class, and hence it is bounded in the Hilbert-Schmidt topology By the Banach-Alaoglu theorem for the Hilbert-Schmidt space, up to a subsequence e —> 0+ we have (1 - A)1/27e(l - A)1/2 - (1 - A)1/27o(l - A)1/2 weakly in the Hilbert-Schmidt topology, namely Tr(K(l-A)1/27e(l-A)1/2) ->■ Tr(K(l-A)1/27o(l-A)1/2), MK Hilbert-Schmidt operators. for a non-negative trace class operator 70 > 0 on L2(lRd) (exercise). From this weak conver- 4.5. CONVERGENCE TO HARTREE MINIMIZER 105 gence and the fact that plE —> \uq\2 strongly in L1(Md), we deduce that pl0 = \uq\2 (exercise). Let us determine the limit 70. Since % —^ 70 weakly in Hilbert-Schmidt and /imf > 0, by Fatou's lemma we have 0 < Tr(/imf7o) < liminf Tr(/imf7e) ->■ 0. E->0+ Thus /imf7o = 0. Since /imf has a unique ground state u0, we have 70 = \\u0){uq\ for some A > 0. But we have proved that p10 = \u0\2, hence A = 1. Thus Is 7o = \uo){uo\ weakly in the Hilbert-Schmidt topology. Consequently (u0,j£u0) = Tr (\u0)(u0\j£^ ->■ 1. This completes the proof. q.e.d. Exercise. Let {An}n>i be a sequence Hilbert-Schmidt operators on L2(M.d). Prove that An —^ Ao weakly in the Hilbert-Schmidt topology if and only if An(-, •) —^ Ao(-, •) weakly in L2(Rd x Rd), where An(x,y) is the kernel of An. Exercise. Let {/yn}n>i be a sequence of Hilbert-Schmidt operators on L2(Rd) such that 7„ > 0 and 7„ —^ 70 weakly in the Hilbert-Schmidt topology. Prove that 70 > 0 and for any self-adjoint operator A > 0 on L2(M.d), we have Tr(^7o) < lim inf Tr(^47„). Here Ti(Aln) := Tr(A1/2~fnA1/2) = Tr^A^J2) 6 [0, 00]. \ Exercise. Let {/yn}n>i be a sequence of trace class operators on L2(R.d) such that^fn > 0, Tr7„ = 1 and Tr((l - A)1/27„(l - A)1/2) < C. 106 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION 1. Prove that up to a subsequence, we have the weak convergence (1 - A)1/27„(l - A)1/2 - (1 - A)1/27o(l - A)1/2 in the Hilbert-Schmidt topology, where 70 > 0 is a trace class operator on L2(Rd) 2. Prove that ioo(Tn>d\ lim / UPln= [ UpJ0, VUeCZ n^oo JRd JRd Hint: If d < 3 you can show that (1 - A)-72£/(l - A)"1/2 is a Hilbert-Schmidt operator on L2(Rd). For general case d > 1, you may use the weak-* convergence in trace class. 4.6 Short-range interactions So far we have derived Hartree theory with regular interaction potentials. Now we consider the case of short-range potentials. Fix a constant (3 > 0 and consider the Hamiltonian N 1 HN = Y(-AXi+V(xij) + --—- £ N^w(N^(Xi - Xj)) i=l KkjXW on L2(M.d)®sN. By restricting to the uncorrelated states u®N and taking the formal limit Ndpw{Npx) -± b50( x), b = l w we obtain the Hartree/Gross-Pitaevskii functional SGP(u)= f (\Vu(x)\2 + V(x)\u(x)\2 + ±\u(x)\4 We consider the ground state energy of Hn EN:=_u inf {^,HN^) and the Gross-Pitaevskii energy 6gp := inf ^gp(^)- IMlL2(Rd)-l 11*1^2(^)8^-1 4.6. SHORT-RANGE INTERACTIONS 107 When Vix) —> oo as |x| —> oo and b > 0, ecp has a unique minimizer (up to a phase) u$ > 0 which solves the Gross-Pitaevskii equation (-A + V+ b\u0\2 - ii)u0 = 0, ii ER. Theorem (Convergence to Gross-Pitaevskii theory). Assume that • 0 < V G A?cM, hm|xKco V(x) = oo, • 0 < w G Cc°°(Md). Fix 1 < d < 3 and 0 < (3 < 1/d. Then we have ,. -EW lim — = eGp. N^oo N Moreover, if^n is an approximate ground state for Hn, namely {^n, Hn^n) = Necp-o(N), then we have the complete Bose-Einstein condensation lim K7£S) = L N^oo N Proof. Step 1. Denote w^(x) = Nb^w(N^x) and define the N-dependent Hartree energy eH,N := inf Sh,n{u) IMIL2(Rd)-l where £h,n(u) := / (|V-u(x)|2 + "\/(x)|-u(x)|2)dx + - // \u(x)\2\u(y)\2WN(x — y)dxdy. JWLxdx = w(N~ßk). Therefore, since w G L (M. ) and f3 < 1/d we have €TN\= \w(N~ßk)\dk = Ndß \w\ = o(N) Therefore, we can repeat the proof of the convergence to Hartree energy to obtain En iim N &h,n = 0. Step 2. Next, we show that Since w > 0 by the Cauchy-Schwarz inequality we have lim eHN = eGP. |-u(x)|2|-u(y)|2K;7v(x — y)dxdy < ulx + u ■wN(x - y) = H^tvIIl1 / M4 = b / \u\4. Therefore, Sh,n{u) < ^gp(^), G H\Rd) and hence by the variational principle we get the uniform upper bound eu,n < eGP, ViV. Moreover, £gp(u) - £h,n(u) = b / \u\4 - // \u(x)\2\u(y)\2wN(x - y)dxdy |-u(x)|2(|-u(x)|2 — \u(y)\2)wn(x — y)dxdy Using l-ufx)!2 — \u("m2 ^-\u(y + t(x-y))\2dt o dt 4.6. SHORT-RANGE INTERACTIONS 109 < / 2\u(y + t(x — y)||V-u(y + t(x — y))\.\x — y\dt Jo and Holder's inequality we find that £gp(^) - £h,n(u) <2 / ( // \u(x)\2\u(y + t(x — y))\\Vu(y + t(x — y))\.\x — y\w]y(x — y)dxdy)dt -1 = 2 1 I (I \u(y + z)\2\u(y + tz)\\Vu(y + tz)\dy)\z\wN(z)dzdt o Jm.d l < 2 1/3 n(y + z)|6dyr~( / Ky + te)|6dyy/6( / \Vu(y + tz)\2dyY'\z\wN(z)dzdt <2 ||ii|||6||Vii||L2 / \z\wN(z)dzdt 0 we have C>£hAun)> [ \VuN\2. Thus {un}n>i is bounded in i/1(lRd). Therefore, £gp(un) — £h,n(un) < CN ^H^atH^-i —> 0 as N —> oo. Consequently, ggp < £gp(un) < £h,n(un) + 0(1)^00 < eH^N + o(l)N_ Thus we conclude that lim eHN = eGP. N=oo Combining with the result from Step 1, we obtain the energy convergence lim — = eGp. AT->oo N Step 3. Finally we prove the BEC. This can be done by the Hellmann-Feynman argument 110 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION again. For any e > 0, we define the perturbed Hamiltonian n HNiE = HN + e^PXs, P=\u0){u0\ i=i and call En,s tbe ground state energy of Hn,s- Then following the above proof of EN = NeGP + o(N), we also have ENtS = NeGPiE + o(N) where inf { Tr((-A + V + eP)7) + - / p'(x)dr). m L2(Kd) I- 2 JRd J 7>0 on Tr 7=1 Therefore, if is an approximate ground state for Hn, then by the variational principle, we have (1) \ 1 n 1 n {UQ, Jyluo) 1 yN, ^2 P^N) = ({$N, ^ HN,eVN) - {^N, HN^N)^j i=i i=i - iTV (Ew'£ ~En + o(iV))= iTV " Ne°p+o(iV)) eGP'££~e°p- Thus to obtain the complete BEC it remains to show that ,. eGP,£ _ eGP -, Iim - = 1. e^0+ e Since eTr(P7) < e we get the uniform upper bound eGP,e ~ eGP < ^ e ~ For the lower bound, let 7e be an approximate minimizer for eGp,e, namely b f Tr((-A + V + eP)l£) + - / p2»cb = eGP,£ + o(e). Then using Tr(P7) > 0 and the Hoffmann-Ostenhof inequality we get £gp(y/A^D < eGp,e + o(e) < eGP + 0(e). 4.6. SHORT-RANGE INTERACTIONS 111 Thus y/pT7 is a minimizing sequence for ecp, and hence we deduce that plE —> uq strongly in L2 (IRd). Since yj~p^~ is bounded in H1 (Rd), by Sobolev's embedding theorem we get yj~p^~ —> u0 strongly in Lp(Rd) for all p G [2, 2*). When d < 3, we get 2* > 6 > 4, and hence o/ p2lE(x)dx = b plE{x)\u0{x)\2dx - - / \u0(x)\4dx + o(l)e_>0 = bTi(\u0\2%) + eGp - p Here ytz G R is the chemical potential in the GP equation (-A + V + b\u0\2 - p)u0 = 0. Thus we find that = Tr((-A + V + b\u0\2 - p)l£) + eGP + o(l)e_>0. Since we have prove eGp,e —> cqp, we get Tr((-A + V + b\u0\2-p)-f£) ->0. Note that -uo is the unique ground state of the operator h = -A + V + %0|2 - A4-Moreover, since V(x) —> oo as |x| —>■ oo, /i has compact resolvent. Thus h has eigenvalues 0 = Ax(/i) < \2{h) < ... Using the spectral gap A2(/i) > \i(h), we conclude that Tr(P7e) 1. which is equivalent to 112 CHAPTER 4. VALIDITY OF HARTREE APPROXIMATION Thus b f eGP,£ + o{e) = Tr((-A + V + eP)le) + - / p\ (x)dx JWLd b f >Tr((-A + V07e) + - / Pl(x)dx + e JWLd > £gv(\J7h) + e > eGp + e which gives the desired lower bound lim inf egp,e — egp > 1. e This completes the proof of the BEC. q.e.d. Remarks: • The same result holds true for all 0 < (3 < oo if d = 1, 2 and all 0 < (3 < 1 if d = 3, but the proof is more complicated. The case (3 > 1/d is interesting because in this case, the range of the interaction potential N~^ is much smaller than the mean-distance between particles iV_1/d. This is called the dilute regime. In contrast, when (3 < 1/d, then the range of the interaction potential N~^ is much bigger than the mean-distance between particles iV_1/d, and hence each particle interacts with many others. This is the reason why the case /3 < 1/d is easier to justify the mean-field approximation (which is somewhat similar to the law of large numbers in probability theory). • In the case d = 3 and (3 = 1, the result is still correct provided that in the Gross-Pitaevskii functional This variational problem has a unique minimizer 0 < / < 1 and it solves the zero-scattering equation £gpW = / ( Vu{x)\2 + V{x)\u{x)\2 + ^\u{x)\2^j the constant b is not j w but rather given by the scattering energy of w (-2A + w)f = 0. 4.6. SHORT-RANGE INTERACTIONS 113 Moreover, we have and 1 , a = —o 8tt is called the scattering length of w. If w is the hard sphere potential of -8(0, R), then a = R. In general, we have Born's series b = 8ira = / wf = / w — w(—2A + w)~1w = ... Thus -g- j w is the first Born's approximation for the scattering length (it is > a except when w = 0). By scaling, the scattering length of N2w(N-) is a/N. The derivation of the GP theory in this critical case is significantly more difficult. We will come back to this problem later when we have more tools from the Fock space formalism and Bogoliubov's approximation. Chapter 5 Fock space formalism Definition. Let ffl be a one-particle Hilbert space. The bosonic Fock space associated to Jrf? is the Hilbert space oo n=0 • Any vector in T has the form \I> = (^„)^L0 where ^n G Jf?®3™ and oo n=0 • The vector Cl = (1, 0, 0,...) is called the vacuum. • The expectation of the number of particles in the state \I> = (x&ra)^L0 £ J7 is oo n=0 This is the same to A/M/) where oo M := y ^ nt jf?®sn n=0 is called the number operator. In particular, (ft, AT ft) = 0. 114 5.1. CREATION AND ANNIHILATION OPERATORS 115 5.1 Creation and annihilation operators Definition. Let Jrf? be a one-particle Hilbert space and let T = J7(J$?) be the bosonic Fock space. For any f 6 Jrf? we can define the creation operator a*(/) and the annihilation operator a(f) on T as follows: • a*(f) : ->■ J^®°n+1 for all n = 0,1, 2,... n+l (a*(f)^n)(x1,... , xn+1) = f(xj)1^n(x1,... , Xj-x, xj+1,. .., xn+1). a(f) : ->■ J^^-1 for all n = 0,1, 2,... (with convention JT^"1 = {0}) (a(/)^n)(xi,. .. = J f{xn)^>(x1,. .. ,xn)dxn. Here we think of Jf? C L2(M.d) to simplify the notation. Remarks: • a*(/)fi = / and a(/)fi = 0. • f i—y a*{f) is linear, but / i—y a(f) is anti-llinear. Example: If J^f is one-dimensional, J^f = span{/}, ||/|| = 1. Then J7(J$?) has an orthonor-mal basis {|n) j-^=o,i,2,... where |0) = (1, 0, 0,...) = tt, |1> = (0, /, 0,...), |2> = (0, 0, /®2, 0,...), \n) = {0,f®n,0,...) In this case, a*(f)\n) = Vn + T\n + l), n>0 a(f)\n) = \/n\n — 1), Vn > 1. 116 CHAPTER 5. FOCK SPACE FORMALISM Exercise. This problem allows us to think of L2 (R) as the Fock space .F(C). Define the operators a and a+ on L2(R) by 1 / d \ . If d t -I--n* = - - a y/2~\ dx)' a \/2~\ dxJ' Define the functions {fn}n>o C L2(R) by Mx) = tt-VVN2/2, /n+1= Vn>0. Vn + 1 1. Prove that [a, a*] = 1 (identity). 2. Prove that afo = 0 and afn+i = y/n-\- lfn for all n > 0. 3. Prove that {fn}n>o is an orthonormal basis for L2(R). Hint: You can use the fact that Span{jo(x)e_x2/2 | p{x) is a polynomial} is dense in L2(R). Exercise. Consider the Fock space J-'^Jff). Prove that for all f G J$?, we have MfMr < \\f\UW1/2n, V* G Q(N). Here Q(Af) is the quadratic form domain for the number operator Af. Exercise. Consider the Fock space J-'^Jff). Prove that for all f 6 J$?, a(f) and a*(f) are adjoints, namely {a(f)V, = a*(/)$)^, W, $ G Q(Af). Theorem (Canonical Commutation Relations - CCR). Consider the Fock space J7(J$?). For all f,g£ Jff, we have [a(f),a(g)] = 0, [a* (f), a* (g)] = 0, [a(f),a*(g)] = (f,g),^. 5.1. CREATION AND ANNIHILATION OPERATORS 117 Here [A, B] := AB - BA. J Proof. We may think of Jf? C L2(Rd) for simplicity Step 1. First, let us prove that [a(f), a(g)] = 0, namely a(.f)a(g) = a(g)a(f)- It suffices to show that a{f)a(g)Vn = a{g)a{f)Vr for any function tyn G Jj?®3"1 and for any n > 2. By the definition of the annihilation operator, we have {a{f)a{g)^n){xi, ...,xn_2) = {a(f)(a(g)^n))(x1, ...,xn_2) = (a(/) (Vii ■■■iVn-i) H> \fn J g{xn)^N{yu...,yn_uxn)dxn ) (x±, xn_2) Q{ß^n) ^11 (*^1 j ' " j %n—l j ^n)d^77,^ dxn — \ = yfn{n - 1) // f(xn_1)g(xn)^n(x1,...,xn_1,xn)dxn_1dxn. Using Fubini's theorem and the bosonic symmetry we can write {a{g)a{f)^n){xi, ...,x n-2 y/n(n-l) J J g( \Jn{n - 1) {{g{xn)f{xn_x)^n{xx, ...,xn,xn_x)dxn_xdxTl (a(/)a(^)^n)(xi,x„_2). Thus a(/)a(^) = a(^)a(/). Step 2. Since a*(f) is the adjoint of a(f), using [a(/), a(<7)] = 0 we have 0 = ([a(f),a(gW = (a(f)a(g) - a{g)a{f))* = (a*(g)a*(f) - a*(f)a*(gj) = -[a*(f),a*(g)]. 118 CHAPTER 5. FOCK SPACE FORMALISM Thus [a*(/),a*(<7)] = 0. Step 3. Finally, we prove that [a(f), a*(g)] = a(f)a*(g) - a*(g)a(f) = (f, g). When testing with the vacuum, we have a(f)a*(g)n - a*(g)a(f)n = a(f)g -0={f,g). Now consider any function tyn G Ji?®an with any n > 1. We have (a(f)a*(g)xifn)(x1, xn) = (a(f)(a*(g)ty n) I 1^1i " 'i xn) ^ n+1 t(^) i—i—t £9{yi)Vn{yi, yi-i, yi+i, yn+i)) ixi, ...,xn) v 1=1 /n+1 f(xn+1) ^gixA^nixx, ...,Xj_i,xm, ...,xn+1)dxn+1 1=1 n „ ) + /^g(Xi) / f(xn+l)^n(xl, ... )dxn+1. On the other hand, (a*(g)a(f)x&n)(xi,xn) = ^a*((/)(a(/)xI/Jj))^ ( = [a*(g)y/n J f{yn)^n{y1,...,yn)dyn^(x1,...,xri) n „ = ^2g(xl) / f(yn)^n(x1,...,Xi_1,xi+1,...,xn,yn)dyn i=i •> n » = y ] 9Jxi) / f{xn+l)^n{xli ■■■ixi-lixi+li ■■■ixnixn+l)dXn+l- i=l J Here in the last identity we simply "renamed" yn to xn+i. Thus a(f)a*(g)Vn - a*(g)a(f)Vn = i. Let an = aiun) on the bosonic Fock space J7(J^f). Prove that J7(J^f) has an orthonormal basis with vectors \ni,n2,...) := (n1\n2\...)-1/2(a*ir(a*2)n\.n. Here n±, n2,... 6 {0,1, 2,...} and there are only finitely many of {n^} are non-zero. Remark: Sometimes it is also convenient to write |0) = £1. 5.2 Second quantization Using the creation and annihilation operators, we can represent many operators on Fock space in a convenient way / Theorem (Second quantization of one-body operators). Let h be a self-adjoint operator on the one-body Hilbert space Jrff. Then the operator on the bosonic Fock space J7(J^f) oo n dT(h) : = 0 ( Y h*) = 0 © h © (h © 1 + 1 © h) © - n=0 i=l is called the second quantization of h. It can be rewritten as dr(^) = Yj hun)a*man- m,n>l Here {un}n>i is an orthogonal basis for Jrf? and an = aiun). The representation is independent of the choice of the basis (provided that all (um, hun) are finite). The identity can be made rigorous at least on the domain oo M U 0% + ... + McJ. M=l N=0 Example: When h = 1 (the identity) we obtain the number operator M = dr(l) = ^ (um,un)a*man = ^ Sm=na*man = ^a*a„- m,n>l m,n>l n>l 120 CHAPTER 5. FOCK SPACE FORMALISM Proof. It suffices to prove that N N i=l m,n>l for all G Ji?®aN and for all N. Recall from a previous computation N r_ {a*man^N){xu ...,xN) = Yjum{xi) j un{y)^n{xu xi+1, ...,xN,y)dy. i=i •> Therefore, Yj(umi hun)(a*man^N)(x1, ...,xN) N r_ = Y/~2(z~2(Um,hUn)Urn(Xl^) / li™(y)^Ar(xi' ^-1,^+1,-,XN,y)&y i=l n m N r_ y)dy i=l n N = YjYj [(\hun)(Un\)x^N (xi,... i=l n N , x^ —1, Xi, x^+l, ..., X_/Y J i=l ra at = ^ [/^at] (xi, ...,xat). i=i Here we have used the Parseval's identity Yj(u™, hun)um = hun m and the the resolution of the identity operator Yj \Un)(Un\ = 1 n (both use the fact that {un} is an orthonormal basis for Jif). This completes the proof. 5.2. SECOND QUANTIZATION 121 q.e.d. Theorem (Second quantization of two-body operators). Let W be a self-adjoint operator on Jif®2 such that W\2 = W2\. Then the operator on the bosonic Fock space CO ©( Yl Wij) =0®0®W®(W12 + W23 + W13)®... n=0 1<^i is an orthogonal basis for Jrf? and an = aiun). The representation is independent of the choice of the basis. Proof. It suffices to prove that Wup uq)a*ma*napaq^N for all $tvi £ ^®>aN and for all N. Recall from a previous computation {a{f)a{g)^N){x1,...,xN_2) = y/N{N - 1) J J f{y)g{z)VN{xu xN_2, y, z)dydz. Therefore, $at, Y {um®un,Wup®uq)a*ma*napaq^N = Yj (Um ® Uni WUP ® ^ (a™an$N, apaq^N = ^ (ira <8> VF-Up (8) uq) / (ama„$Ar)(xi, ...,xAr_2)(apagxI/Ar)(a;i, ...,xAr_2)dxi...dxAr_ ^ <«m ® W% (8) tig) / (y/N(N - 1) // ^(y'^n^O^Jv^ij-j^Jv^jy'^Ody'dz'^x x ^NiN - 1) J J up{y)uq{z)^ N{xu ...,xN_2,y, z)dydz^dx1...dxN_2 Wupn/ Wup ® uq){um ® un)(y', z)up(y)uq(z)x m,n,p,q x $jv(xi, %n-2, y', z')^n(xi, %n-2, y, z)dxi...dxN_2dydy/dzdz/ N{N-1) f Y(Wup ® UM> z')up{y)uq{z)x p,q x $jv(xi, %n-2, y', z')^n(xi, y, z)dxi...dxN_2dydy'dzdz = N{N-T)J2($N, (w\up®uq){up®uq\) VN) p,q = N(N-l)($N,WN-ltN*N)=2($N, Y w^n). lF®sN C with 3V = L2(Rd). Prove that the one-body density matrix 7$ defined by (g,^f)^ = (*,a*(f)a(g)*), Vf,g£J? is the same to the operator defined via the kernel 7$(x, y) = N / ^(x, X2, XN)^(y, X2, XN)dx2---dxN. If ^ G J7(J$?) does not have a fixed particle number, then it is also important to know {y,a(f)a(g)ty) and (\I>, a*(f)a*(g)ty). This gives rise to an operator : Jf* ->■ Jjf. Definition. Let ffl be a Hilbert space and let Jrff* be its dual (i.e. the space of all continuous linear functionals from Jrf? to C). Define the mapping J : Jrf? —> Jrff* by J(f)(g) = {f,g), Vf,geJT- Note that J is anti-linear. The adjoint J* : Jrff* —> Jrf? is an anti-linear map defined by {J*u, v)= {Jv, u)jt?* = (u, Jv)», Vu G ■ '// . v G Jff. By Riesz representation Theorem, J is an anti-unitary, namely ■1.1 I •/•/• I,/-. 5.3. GENERALIZED ONE-BODY DENSITY MATRICES 125 In particular, {Ju,Jv)jt* = {v,u)Jt = {u,v)Jt, \/u,veJ$?. Remarks: • The point here is that we do not identify Jj? to // . but rather think of Jjf* = JJj? with an anti-unitary J. • If J{? = L2(IRd), then we can simply take J as the complex conjugation. \ Definition. Let ^ be a normalized vector in the bosonic Fock space J7(J^') with J\fty) oo. We define the pairing operator : J^f* —> J^f by (g, a*Jf) = Jrff* is defined by (a%g, Jf)*. = {g,a*Jf)jr = (V, a(f)a(g)V), V/, g G 3V. Note that = Ja^J. Remarks: • The advantage of introducing the anti-linear isomorphism J : Jrf? —> Jrff* is that and a\, are linear maps. • The relation a*^ = Ja^J can be seen from the definition of a*^ and the CCR: (a%g, Jf)*. = {*,a(f)*(9)*) = {*, a(g)a(f)V) = {f,ayJg)jr = {JayJg,Jf)jr*, Vf,g£^- • The relation = Ja^J is equivalent to the fact that the kernel a$(-, •) of a$ is symmetric. We can think of J^f = L2(IRd) for simplicity where the kernel a$(-, •) of a$ is defined as (ayjf)(x) = [a^(x,y)(Jf)(y)dy, V/ G L2(Rd). Then by the definition of a$, we can write {g ® /,«*(■,■)) = // g(x)f(y)a9(x,y)6xdy = / g(x)(a9jf)(x)6x 126 CHAPTER 5. FOCK SPACE FORMALISM = {g,a*Jf) = {*,a(f)a(g)V), Vf,gE L (R ) In particular, since a(f)a(g) = a(g)a(f) by the CCR, we deduce that {g® /,«*(■,■)) = {f®9, «*(•,•)) and hence the kernel a$(-, •) is symmetric, i.e. an element of Ji?®a2. Definition. Let ^ be a normalized vector in the bosonic Fock space ^(Jrf?) with Af^) -oo. We define the generalized one-body density matrix T$ as an operator on Jrf? © Jrf?* by the block matrix form := 7* a* a$ \ / 7* a* ■ Jj^J* j \JayJ 1 + Jj^J* Theorem. Let \I> be a normalized vector in the bosonic Fock space ^(Jrf?) with {^jAf^) < oo. Then > 0 on Jrf? © Jrff*. This is equivalently to the operator inequality Consequently, 7* > ®je* = f 7* J(/y \Ja$J 1 + J7$ J*y \J(/ (/, 7*/)^r + (/, ayJg)j? + JayJf)^* + (1 + JiyJ*)Jg)w {f, 7*/)^r + 7*#)^r + a(g)a(f)V) + 0. > 0 5.3. GENERALIZED ONE-BODY DENSITY MATRICES 127 Step 2. From the above proof, we can also see that T$ > 0 is equivalent to (/,7f/) + (?,(l + 7*)ff) >2$t(g,ayJf), Vf,geJfr. By replacing / i—>■ \f and optimizing over A g C, we get <<7,(l+7*)<7> > \(g,^Jf)\2, \/f,gejr. Replacing g by (1 + 7$)_1(/ we get the equivalent formula |(?,(l+7f)"VJ/)|2, Vf,geJf. Then choosing (/ = a^Jf we find that > <«*.//, (i + 7*)~W/) = J*a4(l + 7*)_1a* J on J^. Step 3. Reversely, let us start from the operator inequality 7* > J*a.y(l + 7$)_1a$ J on Jff. Then (<**Jf, (i + 7*)"Wj/> = ll(i + 7*r1/W/||2, v/ g JT. Therefore, by the Cauchy-Schwarz inequality we can bound (fn*f)(g, (i + 7*) > ll(i + 7*r1/W/||2||(i + 7*)-1/2^ll2 > |((1 + 7*)-VV (1 + 7*r1/W/)|2 = K<7, (1 + 7*)" W/)|2, V/, g g ^ which is equivalent to T$ > 0. q.e.d. The above Theorem gives rise to a natural question: given an operator on Jif © J^* of the 128 CHAPTER 5. FOCK SPACE FORMALISM block matrix form r:=f7 a ) \a* i + J7J7 satisfying that V > 0 and a* = JaJ, Tr7 < 00. Then is V the generalized one-body density matrix of a state on the bosonic Fock space J7(J$?)? The answer is yes, provided that we extend the consideration to mixed states. Definition. Let G be a mixed state on the bosonic Fock space T = ^(Jif), namely G > 0 on T and Tij^G = 1, with Tr j?(J\fG) < 00. We define the generalized one-body density matrix of G as an operator on Jif © Jif* of the block matrix form rG := IG OiG a*G l + JlGJ% where 7^ : ,3^ —> Jif and aG : Jif* —> Jif are linear maps defined by (g, lGf)x> = Tr(a*(f)a(g)G), (g, aGJf)^ = Tr (a(f)a(g)G), V/, g G Jif. In case G = |XI/)(XI/| for a normalized vector \I> G J7(J$?), we say that G is a pure state. All of the results discussed above for pure states extend to mixed states, in particular rG>0, a*G = JaGJ, Tr7G = Tr^(A/"G) < 00. We will prove that any such a block-matrix operators on Jif © Jif* is a one-body density matrix of a mixed state. Moreover, the mixed state can be chosen in a special class called quasi-free states. 5.4 Coherent/Gaussian/Quasi-free states In this section we introduce some special states on Fock space which are relevant to the analysis of the Bose-Einstein condensation and fluctuations around the condensate. First, we consider coherent states of the form v'nl n>0 e-H/H2/2ea*(/)Q = e-||/H2/2 0 This is the analogue of Hartree states u®N on Fock space. Similar to Hartree states, coherent 5.4. COHERENT/GAUSSIAN/QUASI-FREE STATES states can be used to describe the Bose-Einstein condensate. 129 Definition. Let Jt? be a one-body Hilbert space. For every f G Jt? (not necessarily normalized), we define the Weyl operator W(f) as a unitary operator on the bosonic Fock space T = T(J??) by W(f)=exp(a*(f)-a(f)). Then W(f)tt is called a coherent state. Theorem. For every f G Jť, the Weyl operator W(f) on J7(Jí?) satisfies W*(f)a(g)W(f) = a(g) + {g,f), W*(f)a*(g)W(f) = a*(g) + {f,g), Mg G Jť. Proof. We will use a "Gronwall argument". In general, if we have two operator A and B, then jt(e~tABetA) = e~tA(—AB + BA)etA = e~tA[B, A]etA. Therefore, integrating over t G [0,1] we find that e~ABeA - B = f e~tA[B, A]etAdt. Jo Now we apply this identity to A = a*(f) — a(f) and B = a(g). By the CCR, [B,A] = [a(g),a*(f)-a(f)] = (g,f) and hence e-tA[B,A]etA = (g,f)e-tAetA = (g,f). Therefore, W*(f)a(g)W(f) - a(g) = e~ABeA - B = f e~tA[B, A]etAdt = (g, f). Jo By the adjointness, it is equivalent to W*(f)a*(g)W(f) - a*(g) = (gj) = (f,g). 130 CHAPTER 5. FOCK SPACE FORMALISM q.e.d. Exercise. Let f 6 ffl and consider Weyl operator W(f) on J7(J$?). Prove that the corresponding coherent state is * := W(f)Q = e-imi2/2e°'(/)fi = e-ll/ll2/2 0 n>0 Prove that Next, we consider excited particles outside of the condensate. We will focus on the quasi-free states, where the excited particles come in pairs. The simplest examples of quasi-free states are Gaussian states Theorem (Gaussian states). Let h > 0 be self-adjoint on Ji? such that Tr(e h) < oo. Then we have the following properties. • The partition function is Z := Tre-dr(ft) = exp ( - Tr(log(l - e~h))^j 6 (0, oo). Consequently, the Gaussian state G = Z~1e~dT(~h^ is well-defined. • The one-body density matrix of G is 1 1G = ~ - 1 This is a non-negative trace class operator on ,3^, namely Tr(A/*G) < oo. The Gaussian state G satisfies Wick's Theorem, namely Tr(af...a*m_1G) = 0, Vm > 1 and Tr(a* ...afmG) = ^ Tr(a#1}a#2)G)... Tr(a#2m_1}a#2m)G), Vm > 1. 5.4. COHERENT/GAUSSIAN/QUASI-FREE STATES 131 Here a# is either a(fn) or a*(fn) with arbitrary vectors (/i, f2,...) C Jff. The set of pairings P2m is P2m = {l with an orthonormal basis {un}n>i for Jrf? and 0 < Ai < \2 < ... with Ye~Xn = Tr(e_fe) < °°- n>l Then we can write dT(h) = ^ \ndr(\un)(un\) = ^ \na*nan n>l n>l where an = aiun). Since a*nan and a*mam commute, we can decompose e-dT(h) _ e- J2n xna*nan _ TT g-A„a>„ Next, recall that the bosonic Fock space J7(J$?) has an orthonormal basis \ni,n2,...) := (n1\n2\...)-1/2(a*ir(a*2)n\.n. 132 CHAPTER 5. FOCK SPACE FORMALISM Here n±, n2,... G {0,1, 2,...} and there are only finitely many > 0. Let us compute e-dT^\ni,n2,...) = JJe-A'a.*a'|ni,n2,...). i For every i = 1, 2,... we have Ae-K-i|ni)Tl2)...) = -e-K-i(a*ai)|ni)n2)„.) = -nie-A 0. Here we used the fact that n2,...) has exactly nt particles in the mode -Uj. Integrating over A G [0, Aj] gives e-x'i e~Xin% and hence rii=0,l,2,... i>l i>l n=0,l,2,... i>l The result can be rewritten in a "fancy way" e-Ai log Z = J] log(l " e"Ai) = Tr(log(l - e~h)) i>l which is equivalent to Z = exp(—Tr(log(l — e h)j). To prove that Z is finite, we need to check na - e-*) > 0, i>l but this follows from the assumption Ei>i e~Xi = Tr e_fe < oo. Exercise. Let {sj}j>i C (0,1). Prove that the following two statements are equivalent. 1- Ei>isi < °o- 5.4. COHERENT/GAUSSIAN/QUASI-FREE STATES 133 [ 2- n>i(i-^)>o-_J Step 2. Now we compute the one-body density matrix 70 Since {un}n>i is an orthonormal basis for J$?, it suffices to prove that {um, lane) = (um, {eh - l)"1^ = Sm=e{eXi - l)"1, Vm, £ > 1. We compute the left side using the definition of 7^ and the fact that \ni,n2,...) are eigen-functions of e_dr^ with eigenvalues Yli>i e~Xi7li. This gives (um,jGUf!) = Tr(a*eamG) = Z'1 Tr {a}ame~AT{h)^ = Z'1 ^ {n1,n2, ...\a*f,ame~dTW\n1,n2, ...) nj=0,l,... = Z~x ^ Y\_e~Xini(niin2, ■■■\a*earn\ni,n2,...) nj=0,l,... i>l Using we can simplify l,jGue)=6m=eZ-1 Y (-^-)lle e i>l A, ( d \ 1 ] V d\,J -1 — e~V e~Xe 1 = $m=ez-T" = &m=l- 1 — e~xt '"" "~ ext — 1' Thus we conclude that jG = {eh —The fact that Tr jc < 00 follows from the assumption Tre"fe < 00 (why?). Step 3. Finally we prove Wick's Theorem. We denote by c% either dj or a* (the indexes i and j may be different). Our aim is to show that Ti[c1c2c3c4...ckG] = Tr[cic2G] Ti[c3c4...ckG] + Tr[cic3G] Tr[c2c4...ckG] + ... + Tr[cicfcG] Tr[c2c3...ck_1G] 134 CHAPTER 5. FOCK SPACE FORMALISM and the result follows immediately by induction. By the same way of computing the partition function and the one-body density matrix, we have Tr[cic2G] = /(ci)[ci,c2] where [cl5 c2] = c\c2 — c2C\ 6 {0, —1,1} and f( ) = { (1 - e^r1 ^ = a„ {Cl> \ (l-e^r1 if c1 = a*. Thus the desired equality is equivalent to Ti[c1c2c3c4...ckG] = f(c1)[c1,c2]Ti[c3c4...ckG] + f{ci)[ci,c3] Ti[c2c4...ckG] + ... + /(ci)[ci,cfc] Ti[c2c3...ck_1G]. Let us focus on the last equality. From the identity c1c2c3c4...ck = [ci, c2]c3c4...ck + ... + c2c4...cfc_i[ci, cfc] + c2c3c4...ckc1 we deduce that Tr [c1c2c3c4...ckG] = Tr [[ci, c2]c3c4...ckG] + ... + Tr [c2c4...ck-i[c1, ck]G] + Tr [c2c3c4...cfcciG]. It is straightforward to see that c\G = e±XjGci where (+) if c\ = a* and (-) if c\ = ar This implies that Tr [c2c3c4...ckc1G] = e±AjTr [c2c3c4...ckGc1] = e±AjTr [c1c2c3c4...ckG] . From that and the definition of / we conclude that Tr [c1c2c3c4...ckG] = ^' ±Aj.Tr [c3c4...ckG] -Tr [c2c4...cfcGJ + ... +--+T-Tr [c2c4...ck_1G\ 1 — e±AJ 1 — e±AJ = /(ci)[ci, c2] Tr[c3c4...cfcG] + /(ci)[ci, c3] Tr[c2c4...cfcG] + ... + /(ci)[ci, cfc] Tr[c2c3...cfc_iG]. 5.4. COHERENT/GAUSSIAN/QUASI-FREE STATES 135 This completes the proof of Wick's theorem. q.e.d. Finally we define \ Definition. Let G be a mixed state on a bosonic Fock space J7(J$?) with Tr(GW") < oo. We call G a quasi-free state if it satisfies Wick's Theorem, namely Tr(af...a*m_1G) = 0, Vm > 1 and Tr(af...a*mG)= ^ Ti{a*(1)a*(2)G)... Tr(a#2m_1)a#2m)G), Vm > 1. Here a* is either a(fn) or a*(fn) with arbitrary vectors (fi, f2,...) C Jt? and P2m is set of pairings P2m = {a e S2m | a(2j - 1) < a(2j + 1), j = 1,... , m - 1, a(2j - 1) < a(2j), j = l,...,m}. • If a quasi-free state is a pure state we call it a pure quasi-free state. • If a quasi-free state G commutes with the number operator, i.e. [G,Af] = 0, we call it a normal quasi-free state. In this case, the paring operator vanishes ac = 0. In principle, any quasi-free state G on J7(J$?) is determined completely by its generalized one-body density matrix rG:=f70 aa ). \a"a 1 + JfcJ'j Moreover, from the general discussion in the previous section we know that Tg > 0 on © ffl*, a*G = JaaJ and Tr^fc < oo. The reverse is also true, namely any block-matrix operator of this form is a Thus a natural question is that given an operator on J/f © J/f* of the block matrix form is the generalized one-body density matrix of a quasi-free state. f-1 136 CHAPTER 5. FOCK SPACE FORMALISM Theorem 5.1. Consider a bounded linear operator on Jrf? © Jrff* r := 7 a .a* 1 + J7J* with r > 0, a* = JaJ, Tij < 00. Then there exists a unique (mixed) quasi-free state G on the bosonic Fock space «F(J^) such that Y = Tq, the generalized one-body density matrix of G. □ The proof of this theorem requires to use Bogoliubov transformations which will be introduced in the next chapter. / \ Exercise. Let AT be the number operator on a bosonic Fock space F^Jff). Let N £ N be a large parameter. 1. Prove that N2 = inf J Tr(M2G) | G a mixed state on T{3^) satisfying Tr(MG) = ivj. 2. Let ^> be an arrbitrary coherent state satisfying {tyjAf^) = N. Prove that (^,A/"2^) = N2 + N. 3. Consider the variational problem EN = inf J Tr(A/"2G) | G = W*(f)KW(f), f £ ^ and K a (mixed) quasi-free state such that Tr(MG) = ivj. Prove that EN = N2 + 0{N2^). Hint: You can write AT (AT— 1) = ^2m n>1 a^a^aman with an = a(un) for an orthonormal basis {un} for Jrf?. You can use the result on the correspondence between G and Tq. Chapter 6 Bogoliubov theory 6.1 Bogoliubov heuristic argument In 1947, Bogoliubov suggested an approximation method to study the low lying spectrum of interacting Bose gases. Recall that the typical iV-body Hamiltonian with pair interactions N HN = Yhi+ }Z Wv i=l KkjXW on Ji?®sN can be extended to be an operator on the bosonic Fock space J7(J$?) as EI = Y hrnna*man + \ W„ 2 m,n>0 m,n,p,q>0 mnpq n P Q where an = aiun) with an orthonormal basis {un}n>o for Jrf? and Bogoliubov suggested that after factoring out the contribution of the Bose-Einstein condensate described by uq, then the contribution from excited particles (orthogonal to uq) can be effectively described by a quadratic Hamiltonian on Fock space ^({uq}^). f-~-) Definition (Bogoliubov's approximation method). 137 138 CHAPTER 6. BOGOLIUBOV THEORY • Step 1 (Ignoring higher order terms) In the second quantization form EI = hmna*man + ^ ^ Wmnpqa*ma*napaq m,n>0 m,n,p,q>0 we ignore all terms with 3 or 4 operators af^0 (a* is either a* or ara/). • Step 2 (c-number substitution) Replacing the operators af by a scalar number y/N^ with N0 > 0. • Step 3 (Cancelation of linear terms) The linear terms containing only one if an/o are cance^ed by the property of uq hu0 0, h := h + N0(W * \u0\2) - fi. • Step 4 (Quadratic approximation) We get HI E0 + HBog rai/i E0 G R and ElBog = Y (hmn + N0Wm0o„)a*man + ^ ^ ^iV0Wmriooaman + ^-c-)- m,n>l m,n>l i/ere u;e write X + h.c. for X+X*. This quadratic Hamiltonian can be exactly diagonalized, leading to an effective description for the spectrum o/HL Explanation: • Motivation of Step 1 (Ignoring higher order terms): most of particles occupy the condensate described by uq, and there are very few particles in the excited modes {un}n^o. Therefore, the contribution from af^0 is much smaller than af, allowing us _LL to ignore terms higher than quadratic in a„/0. • Motivation of Step 2 (c-number substitution): the condensation on the mode u0 implies (aoao) = -/Vq 3> 1 while [aoiao] = 1- Hence we can think of a0 and a*, as they commuted. The most natural candidate for the c-number substitution is thus y/No. Technically, since the term a^aQaoao is quite large, we should rewrite a*0a*0a0a0 = a*0a0(a*0a0 - 1) BOGOLIUBOV HEURISTIC ARGUMENT 139 before applying the c-number substitution. The first two steps result in M « N0h00 + N°(N° ~ 1 Vqooo + V^Vo [(hom + N0W00om)am + h.c. m>l m,n>l m,n>l Step 3 (Cancelation of linear terms) essentially follows from Hartree equation. More precisely from the leading order behavior of uo, we can expect that hu0 0, h := h + N0{W * \u0\2) - p. Consequently for every m/0, h0m + N0Wooom = (um, (h + N0(W * \u0\2)^u0^ (um, iiu0^ = 0. Finally given that the total particle number is N, we can rewrite No = N — N+ where w+ = £>X). n>l Therefore, when think of the mean-field situation VVoooo ~ 1/N we obtain N0hoo + N°(N° ~ 1 Voooo = - A^oo + ^ ~ N+) V000o - ^Woooo iV2 iV A/"2 = A/ioo + — Wbooo - ^(/loo + AVFoooo) - -y VFoooo + -^Wbooo A2 A « A/ioo + — VFoooo - Ooo + AWoooo)^ - — W00oo = Nh00 + —^-r--W00oo - pA+ « E0 + —^--^Wbooo - P 2^ l with N(N - 1) E0 = Nh00 + —^-^VFoooo = AeH. Thus we end up with Step 4 (Quadratic approximation) EI AeH - ii Y a*nan n>l CHAPTER 6. BOGOLIUBOV THEORY + Y (hmn + N^WmOnO + N0WmQQn)a*man + ^ ^ (NoWmnoaa*ma*n + h.c)j m,n>l m,n>l = ^eH + ^ (/w + -/VoWmoon)^^ + ^ X] (NoWmnOoa*ma*n + /l.C.) . m,n>l m,n>l The quadratic Hamiltonian ElBog = ^ (/*mn + -/VoWmOOn)QmQn + \ ^ (iV0Wmrl00aman + ^-C-) • m,n>l m,n>l acts on the excited Fock space ^({mq}"*"). In principle, it can be rewritten as a non-interacting Hamiltonian up to a unitary transformation U on T (called a Bogoliubov transformation), namely iraBogu = eBog + ar(0 with eBog the ground state energy of EIBog and ( > 0 a one-body self-adjoint operator on ,y^+ = {u0}±. Thus in summary, EI ~ Ne^i + MBog = Neu + eBog + Udr(f)U*. The spectrum of the non-interacting Hamiltonian is easy to understand fcefc|efc G 0. In particular, the lowest eigenvalue is inf a(U) w Ne-n + eBog and the ground state is approximately Vfl (after removing the condensation). We will see later that when U is a Bogoliubov transformation, then Vfl is a quasi-free state. 6.1. BOGOLIUBOV HEURISTIC ARGUMENT 141 Remark: Bogoliubov's approximation is a quantized version of Taylor's expansion of the Hartree functional. Recall that if x0 is a local minimizer of a smooth function / : R —> R then near x0 we have Taylor's expansion f(x) = f(xQ) + f(x0)(x - xQ) + ^f"(xQ)(x - x0)2 + o(\x - x0\2) = f(xo) + 7^f'(xo)(x - xo)2 + o(\x - x0\2). Here the first derivative /'(xo) = 0 because the minimizing property of x$. Similarly near the minimizer uq of the Hartree functional (we think of the case J^f = L2(IRd)) £h{u) = (u, hu) + JJ{N - l)W{x - y)\u(x)\2\u(y)\2dxdy under the constraint \\u\\ = 1 we can write for v £ {"^0}^ ( u0 + v \ ^ , N , 1 - (1 + ||^||2)i/2 J = ^h(«o) + -Hess £b(u0)(v, v) + o({v, (h + C)vj). The Hessian operator is ^Hess £n{uQ){v,v) 1 = (v, hv) + - / / w{x - y) ( v(x)u0(x)u0(y)v(y) + v(x)u0(x)u0(y)v(y) + v(x)u0(x)u0(y)v(y) + v(x)u0(x)u0(y)v(y) )dxdy l/fv\fh + K1_K2 \fv\\ 2\\v)\ K*2 h + K[)\v)l Here we identify L2(IRd)* = L2(lRd) and Ki, K2 are operators on L2(lRd) with kernels Kx(x,y) = u0(x)u0(y)(N - l)W(x - y), K2(x,y) = u0(x)u0(y)(N - l)W(x - y). The second quantization form of the Hessian matrix can be obtained by formally replacing vix) by an operator a*ix) which creates an excited particle at x, and v{x) by an operator aix) which annihilates it. This gives HBog := //(h + K1)(x,y)a*xaydxdy + ^ J J (K2(x,y)a*(x)a*(y) + K2(x,y)a(x)a(y)jdxdy. 142 CHAPTER 6. BOGOLIUBOV THEORY Here we are working on the excited Fock space ^({uq}^). This is the same to write m,n>l m,n>l which coincides to y] (hmn + N0Wm00n)a*man + ^ ^ (^ol^mnooa>^ + ^-c) • m,n>l m,n>l up to a small adjustment iVo ~ (iV — 1). 6.2 Example for the homogeneous gas Let us consider the simplest model where we have N bosons in a unit torus Td (i.e. [0, l]d with periodic boundary condition). The particles interacts via an interaction potential W = (N - l)"1^ with w(x) = w(—x) = Y^ w(k)elk'x. fcG27rZd We will assume that the interaction potential is of positive type and smooth, namely 0 < w G £\27rZd). Here we do not put any external potential, and hence the system is translation-invariant. The corresponding iV-body Hamiltonian reads N 1 HN = J^(-AXJ + j^-~jw{xi - Xj) i=i acting on L2(Td)®sN. In this case, the Hartree theory has a unique minimizer (up to a phase) u0(x) = 1, Vx G Td. Exercise. Consider the Hartree functional Sniu) = I \Vu\2 + ]- if w(x - y)\u(x)\2\u(y)\2dxdy. ■JTd ^ J JTdxTd 6.2. EXAMPLE FOR THE HOMOGENEOUS GAS 143 Prove that if 0 < w £ l1{2TTrLd), then the Hartree energy is inf (£H(«) I u £ H\Td), \\u\\L2{Jd) = l} = ±w(0). Moreover, u0 = 1 is the unique Hartree minimizer (uniqueness is up to a phase). Now we apply Bogoliubov's heursitic argument to this Hamiltonian. We take the orthonormal basis {uk} for 3t = L2(Td) with uk(x) = elk-x, Vk £ 2irZd. Then we have h = —A + N0(W * |ii0|2) -fJL = -A (u0 = 1 is the unique ground state for h) and N0Wm00n m (N - l)Wm00n = I um(x)u0(y)w(x - y)u0(x)un(y)dxdy ff e-im-x uj(k)elk Bp > 0, we have Ap (a;ap + a_pa_p) + Bp(a*pa*_p + apa_p) = ^A2p-B2 - Ap + ^ A2p - B2 (b*pbp + b*_pb_ where bp = ap J is2 + l + a*_nvv, vv= J- 1 / Ar. 1 . Moreover, prove that [bp, bq] = 0 and [bp, b*] = 5p=q for every p,q G 27rZd. In particular, applying the above exercise with Ap = \p\2 + w(p) and Bp = w(p) we obtain gb°s = \ z\Z [(l^l2 + ™(P)) (a*Pap + a*-Pa-p) + ™(p)(a*pa*-P + aPQP) 0/pG27rZd = eBog + X] ePb*pbP 0/pG27rZd with eBog — 2 ^ 0/pG27rZd -^j2 Ap) x- y (v\p\4+2\p\2™(p) - \p\2 - ™(p)) 0/pG27rZd and ep = y?\p\A + 2\p\2w(p), bp = apJ u2 + 1 + a*z/p, z/p = 1 / \p\2 + w(p) 2\y/\p\4 + 2\p\2w(p) 1 . 6.3. BOGOLIUBOV TRANSFORMATION 145 In summary, for the homogeneous gas, Bogoliubov's approximation reads HN —w(0) + eBog + 2^ epb*Pbp- 0/pG27rZd Note that fo*'s form new creation/annihilation operators as they satisfy the CCR [bp,bq]=0, [b*p,b*q] = 0, [bp,b*q] = 5p=q, \/p,qE2irZd. So we can treat X]o/pG27rZd epbjfip as ^^e second quantization of a one-body operators. More precisely, we can show that there exists a unitary operator U (the Bogoliubov transformation) on the bosonic Fock space such that V*apV = bp = ap^l + uj + a*_pup, VO + p E 2irZd. Consequently, epb*pbp = V*( ePa>P)u = U*dr( ^ ePK>>Kl)U- 0/pG27rZd 0/pG27rZd 0/pG27rZd whose eigenvalues are Y2 epnpi np = 0,1,2,... 0/pG27rZd Thus the low lying eigenvalues of Hn are of the forms N x ytu(O) + eBog + 2^ ePnP' np = 0,1,2,... 0/pG27rZd This calculation goes back to Bogoliubov's 1947 paper. However, this formula of the excitation spectrum was only proved rigorously in 2010 by Seiringer (CMP 2011). In the homogeneous gas, the diagonalization of the quadratic Hamiltonian HiBog can be done explicitly in the level of 2 x 2 matrices. In order to deal with inhomogeneous trapped cases, it is important to understand Bogoliubov transformations in a more abstract level. This will be done in the next section. 6.3 Bogoliubov transformation 146 CHAPTER 6. BOGOLIUBOV THEORY Definition. A unitary operator U on the bosonic Fock space J7(J$?) is called a Bo-goliubov transformation if • There exist bounded linear maps U : .3^ —> Jif and V : Jif* —> Jif such that for all f G 3tf: V*a*(f)V = a*(Uf) + a(VJf), V*a(f)V = a(Uf) + a*(VJf), Ua*(/)U* = a*{U*f) - a(J*V*f), Ua(/)U* = a(U*f) - a*(J*V*f). • The states Vfl, U*£l has finite particle number expectation {vn,Afvn) < oo, {v*n,Afv*n) < oo. Example (1 dimension): Consider the case dim Jif = 1, i.e. Jif = Span{/}, ||/|| = 1. Then for every A G IR the following mapping Ua = exp is a Bogoliubov transformation on the bosonic Fock space J7(J$?) and UAa*(/)UA = cosh(A)a*(/) + sinh(A)a(/), U^(/)UA = cosh(A)a(/) + sinh(A)a*(/). In fact, since the operator B = \{a\ff-aU)2) is anti-hermitian (B* = —B), the mapping is a well-defined unitary operator. Its action on the creation and annihilation operators can be computed using the Duhamel expansion and the CCR. For example when A > 0 using e~XBa{f)e^ = a{f) + £ jt[e-tBa(f)etB)dt = a{f) + £ e"tB[a(/), B]etBdt 6.3. BOGOLIUBOV TRANSFORMATION 147 and Hf),B] = \(aU)a\ff-a\ffa{f) 2 we can write = \ ( W), a*(f)]a*(f) + a*(f)[a(f), a*(f)] ) = a*(f) x e-"Ba(f)e"B = a(f)+ / e~tB a\f)etBdt. Jo By taking the adjoint, we also obtain e-XBa*(f)eXB = a*(f) + [\~tB a{f)etB dt. Jo Using repeatedly these equalities, we have the series expansion e-XBatf)eXB = a{f) + f\~tB a* {f)etB dt Jo = a(f)+ [ a*(f)dt+ [ [ 1 e-tBa(f)etBdtd\1 ■Jo Jo Jo = a(f)+ [ a*(f)dt+ [ [ 1a(/)dtdA1+ / / * ! " e~tBa*{f)etBdtdX2dXx ■Jo Jo Jo Jo Jo Jo = a(f)+ [ a*(f)dt+ [ [ 1a(/)dtdA1+ f [ * f * a*(f)dtd\2d\1 Jo Jo Jo Jo Jo Jo l>A l>Ai l>A2 I'M / / / / e-tBa(/)etBdtdA3dA2dAi = ... Jo Jo Jo Jo M \2n M \2n+l fX rXi pX2M+i Let us show that the series converges as M —> oo. We have Exercise. Let f G , ||/|| = 1. For every X G R, define A Ua = exp 148 CHAPTER 6. BOGOLIUBOV THEORY Prove the operator inequality on the bosonic Fock space «F(J^) V*x(a*(f)a(f) + a(f)a*(f))Vx < e2^ (a*(f)a(f) + a(f)a*(f) Hint: You can use Gronwall's argument. Thus for any ^ G Q{M) = D(a(f)) we have ||e-íBa(/)eíB^|| = |K/)eíB^|| = {V, e~tB a* {f)^^13^)1'2 < e^l (2||a(/)tf||2 + 1 1/2 and hence Thus in summary / ... / e-tBa{f)etBátá\2M+í...á\ím o Jo Jo < / ... / ||e-íBa(/)eíB^||dídA2M+1...dA1 Jo Jo Jo < C* / / ... / eídídA2M+1...dA1 Jo Jo Jo y2M+2 < C^ex——-- -> 0 as M ->■ oo. ~ (2M + 2)! 00 \2n 00 \2n+l ^ (27 n=0 V eA + e~A «(/) ^ (27 eA - e"A 2 w' 2 = cosh(A)a(/) + sinh(A)a*(/). *(/) Thus UAa(/)UA = cosh(A)a(/) + sinh(A)a*(/), UAa*(/)UA = cosh(A)a*(/) + sinh(A)a(/) where the second identity follows from the first one by the adjointness. Since UA = U_A, we also have the reverse formula UAa(/)UA = cosh(A)a(/) - sinh(A)a*(/), UAa*(/)UA = cosh(A)a*(/) - sinh(A)a(/). Example (2 dimensions): The following example goes back to the original 1947 work of 6.3. BOGOLIUBOV TRANSFORMATION 149 Bogoliubov. Consider the case dimJt? = 2, i.e. J/f = Span{/i,/2} with {fi, fj) = Sl=r For every A G R define where at = a(fl) the annihilation operator. Then Ua is a Bogoliubov transformation on the bosonic Fock space J7(J$?) and These identities can be proved using the Duhamel expansion and the CCR as above. For example, when A > 0 we can write Again, since UA = U_a, we have the reverse formula UaQ±Ua = cosh(A)ai — sinh(A)a2, Ua02Ua = cosh(A)a2 — sinh(A)a*. For future applications, we need to understand the Bogoliubov transformations on J7(J$?) with higher dimensional cases, including the case dim = +oo. A fundamental question is under which conditions on the linear maps U : Jt? —> Jt? and V : Jtf* —> Jt? we can find a Bogoliubov transformation U on J7(J$?) implementing them. The necessary and sufficient conditions on U and V for the existence of a Bogoliubov transformations are given by the following Ua = exp A(a^Q2 — aia2) UaQiUa = cosh(A)ai + sinh(A)a2, UaQ2Ua = cosh(A)a2 + sinh(A)a^. cosh(A)a! + sinh(A)a2- 150 CHAPTER 6. BOGOLIUBOV THEORY Theorem (Existence of Bogoliubov transformations). The bounded linear maps u : Jif —> Jif and v : Jif* —> Jif are implemented by a Bogoliubov transformation U on j7(J$?) if and only if the following conditions hold: • Shale condition: Tt(VV*) < oo. • Symplectic condition: uu* - vv* = i = u*u - j*v*vj, u*vj - (u*vjy = o = vju* - (vju*)*. Remarks: • In the following proof, we will deduce the Shale condition from the identity Tr(VV*) = (UQ,A/UQ) < oo. In fact, even if we define Bogoliubov transformation without requiring {U£l,AfU£l) < oo, then the existence of Bogoliubov transformation always requires TiiVV*) < oo (and hence implies (Vfl,MVfl) < oo automatically). The proof of the latter point is more difficult (we do not need it). • The "symplectic condition" can be written in a compact form with symplectic block matrices on // Jf* v*sv = vsv* = s, where \ JVJ JUJ* / \ 0 -1 In particular, V is invertible. The following exercise tells us that we can deduce one identity vju* = (vju*)* from the others. Exercise. Let u : Jif —> Jif and v : Jif* —> Jif be bounded linear operators such that uu* = i + vv*, u*u = i + j*v*vj, u*vj = {u*vjy. 1. Prove that VJU* = (vju*)*. Hint: This is equivalent to vju*uu* = (vju*)*uu*. 6.3. BOGOLIUBOV TRANSFORMATION 151 2. Prove that VSV* = V*SV = S with V:=( U V ), S-.J1 ° Y \ JVJ JUJ* J \ 0 -1 J 3. Prove that z/Tr(VV*) < oo, then V*V — 1 and W* — 1 are Hilbert-Schmidt operators on // // . Proof of the theorem. We prove the necessity in Steps 1,2 and the sufficiency in Steps 3,4. Part A: Necessity. We assume that there exists a Bogoliubov transformations U associated to U and V. Step 1. We check the Shale condition Ti(VV*) < oo. Let {fn}n>i be an orthonormal basis for J$?. Then n = fa (a.*(Ufn) + a(VJfnj) (a(Ufn) + a*(VJfn^tt n = Y,{n,i and {/i}i>i for J^f such that Uui = cosh(Aj)/j, V' Jui = sinh(Aj)/j, Aj > 0, Vz = 1, 2,... From the symplectic condition we know that the anti-linear operator k = u*vj is Hermitian, 6.3. BOGOLIUBOV TRANSFORMATION 153 i.e. K = K*, and it commutes with U*U: U*UK = U*UU*VJ = U*(1 + VV*)VJ = U*VJ + U*VV*VJ = U*VJ(1 + J*V*VJ) = KU*U. Since U*U — 1 = J*V*VJ is trace class (thanks to the Shale condition), U*U has an or-thonormal basis of eigenvectors. Moreover, since K commutes with U*U, it leaves invariant eigenspaces of U*U. Since K = K* (anti-linear Hermitian) and K*K = J*V*UU*VJ is linear trace class, we can diagonalize further K on each eigenspace of U*U (see an exercise below). t * Exercise. Let K be a bounded anti-linear map on a Hilbert-space . Assume that K = K*, namely {Ku,v) = {u,Kv) = {Kv,u), ^u,v 6 Jff. Moreover, assume that the operator K2 has an orthonormal eigenbasis. Prove that K has an orthonormal eigenbasis with non-negative eigenvalues. Hint: You can write K2 — X2 = (K — X)(K + A). Thus in summary, we can find an orthonormal basis {ut}t>i for Jrf? of joint eigenvectors of U*U and K, namely U*UUi = yU2-Uj, i^-Uj = ^j-Uj, Vz > 1. Here [i% > 1 because U*U > 1 and Aj > 0. Define {/j} by Uui = fiifi, Mi > 1. Then we have 1 which implies that U*

1 has trivial kernel. Thus {/j} is an orthonormal basis for J$?. 154 CHAPTER 6. BOGOLIUBOV THEORY On the other hand, since u% is also eigenfunction of K = U*VJ, we have (/,-, VJu%) = ^{Uuj, VJu%) = ^{uj, U*VJui) = \is ;;A ;. Mi,j. Since {/j} is an orthonormal basis for Jif, we can use Parseval's identity VJu, = VJui)fj = Y I1, ;s,'v, = Vifu v% := p"1^ > 0. 3 3 Thus we have found orthonormal bases {u^^i and {/i}i>i for Jif such that Uui = fiifi, VJu% = i>ifi Mi = 1, 2,... with /jj > 1 and z/j > 0. Moreover, p% = y/l + v2 because Pi-is? = WUu.f - \\VJUl\\2 = {Ul, (U*U - J*V*VJ)Ul) = {uljUl) = 1. Since ji2 — v2 = 1, we can write pt = cosh(Aj) and vl = sinh(Aj) for some Aj > 0. Step 4. Now we want to construct a unitary operator U on J7(J$?) such that U*a(ui)U = a(Uui) + a*(VJui) = cosh(Al)a(/l) + sinh(Al)a*(/l), V* > 1. This looks quite similar to the one-dimensional case that we discussed before, except that in the left side we have a(ui) instead of a(/j). More precisely, from the previous discussion on the one-dimensional case, we know that there exists a unitary operator U on J^(Jif) such that U*a(/l)U = cosh(Al)a(/l) + sinh(Al)a*(/l), V* > 1. In fact, U is given by the explicit formula U = i[exP (^(a*(fl)2 - a(fl)2)) = exp ( £ ±(a*(fl)2 - a(fl)2)). i>l i>l Here in spite of the infinite product, or the infinite sum, the unitary operator U is well-defined. To be precise, the condition Ti(VV*) < oo is equivalent to Yli>i sinh(Aj)2 < oo, 6.3. BOGOLIUBOV TRANSFORMATION 155 which is also equivalent to J^i>i < 00 (wny?)- Consequently, if we define i>l then by the Cauchy-Schwarz inequality we have the operator bound ±iB < I J>* (,/>(/,) + AjaC/Oa'C/O) < (l + £ *?) + ^ = "L i>i i>i This ensures that B is well-defined on D(J\f) (a dense subset of J7 (Jí?)) and it is anti-hermitian (B* = —B). Thus U = eB is a unitary operator on J7(Jť). Then we can choose the desired transformation U as U = YÍLJ where Y is the unitary transformation on J7(Jí?) such that Y*a(Ul)Y = a(fi). The latter unitary operator Y simply corresponds to changing from the orthonormal basis (nxl^LO-^VMrVMr.-A rii = 0,1,2,... to the orthonormal basis (n1\n2\...)-1l2(a%f1)r(a*(f2)r..ni nl = 0,1,2,... Thus we conclude that V*a(Ui)V = V*Y*a(Ui)YV = V*a(fi)V = cosh(Al)a(/l) + sinh(Al)a*(/l) = a(Uut) + a*(VJut), V* > 1. By the linearity, we obtain V*a(u)V = a(Uu) + a*(VJu), VutJť. The inverse of U is also easy to compute. Using the property of the inverse of U (see the 156 CHAPTER 6. BOGOLIUBOV THEORY one-dimensional case) and the definition of Y, we find that Ua(/l)U* = YUa(/i)U*Y* = Y(^ cosh(Al)a(/l) - sinh(Al)JY* = cosh(Aj)a(-Uj) — sinh(Aj)a*(-Uj). From the choice of orthonormal bases {-Uj} and {fz}, we also find that U Ji = "7777—m" = \Wui\K = cosh(Al)-ul, \\UUi\\ J*V*Uui Ku% . J V fi = utt ii = 1177- ii = smh(Al)iil. || U lli || ll^^ill Thus Ua(/i)U* = cosh(Al)a(nl) - sinh(Ai)a*(ui) = a(U*fl) - a*(J*V*fi), Mi > 1 and hence by the linearity Ua(/)U* = a(U*f)-a*(J*V*f), Mf G Jff. Finally it is easy to see that (UQ,A/UQ) = Tr(W*) < oo and a similar bound holds for V*fl. This completes the proof of the existence of the Bogoli-ubov transformation. q.e.d. Let us end this section by a general remark on the one-to-one correspondence between linear maps (U, v) and the set of Bogolliubov transformations (two unitary operators U and zV with z G C, \z\ = 1, are considered the same). /-\ Definition. For a given Hilbert space ,3^, consider the subset of bounded linear operators on .// .// ' i ^1 < 00• 158 CHAPTER 6. BOGOLIUBOV THEORY Consider the state := JJ(l-tanh(Al)2)1/4exp ( tanh(Aj) a*(fi)2)n. 2 Prove that ^Bog is a normalized vector in the bosonic Fock space «F(J^) and (cosh(Al)a(/l) + sinh(Al)a*(/l))QB, 0 V* > 1. J 6.4 Diagonalization of block operators Now we discuss the diagonalization of block-operators on © J^* by symplectic operators in Theorem (Diagonalization of bosonic block operators). Let h : Jif —> Jif and k : ffl* —> Jif be linear operators satisfying • h = h* (h can be unbounded); • k* = JkJ and Tiikk*) < oo; • There exists a constant e0 > 0 such that Then we can find an operator V £ & and a self-adjoint operator £ > 0 on Jif such that The main result of this section is J If dim Jif < oo, the result goes back to Williamson's Theorem (1936). The important case of 2 x 2 real matrices was solved explicitly in Bogoliubov's 1947 paper. This 2x2 case can be generalized easily to: 6.4. DIAGONALIZATION OF BLOCK OPERATORS 159 Example (Commutative case). Let h and k be multiplication operators on Jif = L2(£l,C), for some measure space £1. Then J is simply complex conjugation and we can identify Jif* = Jif for simplicity Assume that h > 0, but k is not necessarily real-valued. Then (h k \ > 0 on .// // . k h J if and only if — 1 < G < 1 with G := \k\h~1. In this case, if we choose '1 1/1 ~G V2 2vT^l 1 then V*^V = with £ := Wl - G2 = Vh2 - k2 > 0. If A > e0 > 0, then /i > e0 > 0. Combining with Ti{kk*) < oo we obtain Tr(GG*) < oo, which is equivalent to Shale's condition for V. Remark: As proved by N-Napiorkowski-Solovej (JFA 2016), the above theorem still holds true if we replace the gap condition A > £o > 0 and the Hilbert-Schmidt condition Tiikk*) by the weaker/optimal conditions: A > 0, Tr^kk*^1) < oo. We will follow the proof of this paper. Our starting point is a "fermionic analogue" of the above theorem. Lemma (Diagonalization of fermionic block operators). Let B be a self-adjoint operator on // // ■ such that Kei(B) = {0} and JBJ = -B, J = 0 J* J 0 Then there exists a unitary operator IA on ,3^ © .3^* such that JIAJ = IA and a self-adjoint operator £ > 0 on ,3^ such that U*BU = £ o o -JiJ* 160 CHAPTER 6. BOGOLIUBOV THEORY Remark: • Note that J is a anti-linear map on © J^* and J = J* = J —1 • Any linear operator on Jif © Jif* of the block form A k* JhJ* h k ) k* = JkJ, h = h* commutes with J, namely JAJ = A. This corresponds to bosonic block operators. On the other hand, in the above lemma we require that B anti-commutes with J, namely JBJ = —B, and this corresponds to fermionic block operators. The difference is that bosonic block operators are diagonalized by symplectic operators, while fermionic block operators can be diagonalized by unitary operators which is easier to deal with. • The result in the above lemma also holds if dimKer(23) is either even or infinite (and we only know £ > 0), but we will not need this extension. Proof of the lemma. Since B is self-adjoint Ker(B) = {0}, by the Spectral Theorem we can decompose P+ :=t{B> 0){JT © JT), P_ :=t(B< 0){JT © JT). The condition JBJ = —B implies that P_ = JP+. Thus we have P+ © JP+ = 3t © 3t* = {Jf? © 0) © J(3t © 0). The latter equality in particular implies that ,3^ © 0 and P+ have the same dimension (finite or +oo). Therefore, there exists a unitary operator W : Jif © 0 —> P+. Then JWJ : J{3if © 0) —> JP+ is also a unitary operator. Consequently, 3t®3t* P+®P- where U :=W®JWJ is a unitary on © J^*. It is also clear from the definition of U that JUJ = U. 6.4. DIAGONALIZATION OF BLOCK OPERATORS 161 It remains to show that U*BU is block-diagonal. Note that for every / G J^, we have W{f © 0) G P+, and hence BW{f © 0) G P+, and then W*BW{f © 0) G Jt? © 0. Thus we can define a linear operator £ : ffl —> ffl by ($/)©0:=^W(/©0), V/G^T. Note that ^ > 0 because = 0 (6.4.1) for all 0 7^ / G J$?. The last inequality follows from the facts that W(f © 0) G P+ and that the restriction of B on P+ is strictly positive. We will now show that WBU which is equivalent to U*BU{f © 0) U*BU(0 © Jf) Indeed, using U{f © 0) = W{f © 0) G P. U*BU{f © 0) = W*BW(f © 0) = (f/) © 0. Similarly, using W(0 © J/) = JWJ{0 © J#) = © 0) G P_, we have BU{0 © J#) = BJW{f © 0) G P- = JP+, and hence U*BU(0 © Jf) = (JW*J)BJW(f © 0) = W*(^^)W(/ © 0) = -JW*BW(f © 0) = -J{{U) © 0) = -J£/. Here we have used JBJ = —B. This completes the proof of the lemma. q.e.d. Proof of the theorem. Since A > 0 is self-adjoint, we can define A1^2 > 0. Let us consider - r 0 V o -JiJ* ) = (£/)©o, = o©(-J£/), v/G^r. _, we have © 0) = BW(f © 0) G P+, and hence £ := A1/2SA1/2. 162 CHAPTER 6. BOGOLIUBOV THEORY It is clear that B is self-adjoint and Ker(23) = {0} because Ker( 0 on Jif such that U*BU = f ^ ° | =: D V o -JiJ* ) Now we define V:=A-1/2\B\1/2U. This choice diagonalizes A because VAV = (U*\B\1/2A-1/2)A(A-1/2\B\1/2U) =U*\B\U = \U*BU\ = \D\=i^ ® J . Boundedness of V. Since , I h k \ A=\ > e0 > 0 \ k* JhJ* and A-sAs=(h k j-[ -M=2 0 M \ k* JhJ* J \ -k* JhJ* J \ k* 0 J is bounded (because k is bounded), there exists S > 0 such that 5A < SAS < 5-xA. Therefore, 5A2 Y > 0, then X1/2 > Y1/2). This follows from the representation 2 p x 6.4. DIAGONALIZATION OF BLOCK OPERATORS 163 (this formula holds for real numbers X > 0, and hence it holds for self-adjoint operators X > 0 by the functional calculus) and the fact that X \—> Xj[X +t2) is operator monotone. Exercise. Let X, Y be two self-adjoint operators on a Hilbert space. Prove that if X > Y > 0, then X~x < Y~\ Hint: You can use the fact that Z*Z < 1 implies ZZ* < 1. Exercise. Prove that for any power s 6 (0,1), the function 0 < t i—> ts is operator monotone, namely if X, Y are two self-adjoint operators on a Hilbert space and X > Y>0, then Xs > Ys. Thus we have proved that \B\ < S~1/2A, which is equivalent to {A~1,2\B\1I2){A~1,2\B\1I2)* = A~1/2\B\A~1/2 < 5-1'2. Consequently A'1?2^2 is well defined on D^1'2) and can be extended to be a bounded operator on Jf 0 J^*. Thus V = A~1/2\B\1/2U is well-defined as a bounded operator on J^f © J^f*. Symplectic condition of V. Indeed, because J commutes with A, \B\ and U, it also commutes with V. Thus V has the form V = U V JVJ JUJ* Moreover, using B = A1/2SA1/2 and U*BU = we find that V*SV = (U*\B\1/2A-1/2)S(A-1/2\B\1/2U) = W\B\1'2{A-1/2SA-1/2)\B\1I2U = W\B\1,2(B-1)\B\1,2U = \U*BU\1/2{U*BU)-1\U*BU\1/2 = IDI^D-^DI 1/2 o \f r1 o \f e/2 o o j^/2j* I I o -Jer1 J* ) V o j^/2j* 164 CHAPTER 6. BOGOLIUBOV THEORY VSV* = (A-1/2\B\1/2U)S(U*\B\1/2A1/2) = A-1I2U\WBU\1/2S\WBU\1/2U*A-1/2 = A-^iUlDl^SlD^U^A-^2 = A~1/2UDU*A~1/2 = A-1,2iUDW)A-1/2 = A~1/2BA~1/2 = A-^iA^SA^A-1'2 = S. Shale condition of V. Finally we prove the Shale condition Tr(V*V) < oo, which is equivalent to VV* - 1 = (A-l,2\B\l,2U)iU*\B\1/2A-1/2) - 1 = A-1/2\B\A~1/2 - 1 = A-1/2(\B\ - A)A~1/2 is a Hilbert-Schmidt operator on Jif © Jif*. Using again the representation of the square root x1'2=- r -*-dt=- r (i - ~^—) dt (6.4.2) it J0 t2 + X ir Jo V t2 + XJ v ' and the resolvent identity t2+A2 t2 + B2 t2+A2K 't2 + B2 we can write VV* - 1 = A~1/2(\B\ -A)A~1/2 2 r ..w,/ i i i A~l/2 ~ A~l/2 t2dt ir Jo \t2 + A2 t2 + B2 - r A~1/2-r-^(®2 - A2)^—A-^2t2dt tt Jo t2 + A2( Jt2 + B2 - l°° A~1/2_-_(A1/2SASA1/2 - A2)_-_A~1/2t 6.5. CHARACTERIZATION OF QUASI-FREE STATES 165 2 r°° Wo t2 + a2 1 (sas - a)a1'2-^—a-1'2 t2dt. tz + Bz Note that E := sas -a = -2 0 k k* 0 is a Hilbert-Schmidt operator on J^f © J^f*. Moreover, using a > Eq > 0 we can bound in the operator norm 1 t2+a2 < 1 op t2 + e2' Combining with 5 a > B > 5a we also have A1'' 1 t2 + B2' a-1'2 = a1'2^-1 op 1/2. 1 t2 + B2 \b\x'2a- < wa1'2^-1 /2| I op 1 t2 + B2 ■1/2 op |S|l/2^-l/2| op op < t2 + 5e20 Therefore, by the triangle inequality for the Hilbert-Schmidt norm, we find that IVV* - l||„s < - 7T 1 E^l/2^_L^^-l/2 < 7T t2+^2 1 t2 + B2 t2dt HS op t2+a2 2 f°° 1 7T \E\ HS a1'2 1 t2 + b2 a-1'2 t2dt op This completes the proof of the theorem. q.e.d. 6.5 Characterization of quasi-free states Recall from the previous chapter that a (mixed) state G on a bosonic Fock space J7(J$?) is a a quasi-free state if Tr(GW') < oo and G satisfies Wick's Theorem, namely Tr(af...a*m_1G) = 0, Vm > 1 and Tr(af...a*mG)= £ M4{i)4{2)G)-Tr(a#2m_1}a#2m)G), Vm > 1. 166 CHAPTER 6. BOGOLIUBOV THEORY A simple but very useful observation is that Bogoliubov transformations leaves invariant the set of quasi-free states. Theorem. Let G be a (mixed) quasi-free state on a bosonic Fock space ^(Jif). Let U be a Bogoliubov transformation on ^(Jif). Then V*GV is a quasi-free state. Proof. Recall the definition of the generalized creation/annihilation operator A(f®Jg) = a(f) + a*(g), Vf,geJ>?. Then Wick's Theorem can be rewritten as Ti(A(F1)...A(F2m_1)G) = 0, and Tr(A(F1)...A(F2m)G) = ^ Tr [A(Fa{1))A(Fa{2))G]... Tr [>l(Ft7(2m_1))>l(Ft7(2m))G for all m > 1, for all vectors /•', { .// .// ' (why?). On the other hand, the Bogoliubov transformation U acts as IL4(F)U* = A(VF), VF £ JT ® JT* for some bounded linear operator V on Jif © Jif*, ( U V \ (l 0 , V = , V*SV = VSV* = S= \ , TrCVV*) < oo. \ JVJ JUJ* J I 0 - 1 Since U is a unitary operator, we have Ti(A(F1)...A(Fn)V*GV) = Tt (VA(F1)...A(Fn)V*G) = Tr (A(VF1)...A(VFn)G) for all n > 1 and for all F G Jif © Jif*. Thus we see immediately that V*GV also satisfies Wick's theorem. Finally, Tr(A/lTGU) = Tr(UA/lTG0 < Tr(C(A^ + 1)G) < oo. 6.5. CHARACTERIZATION OF QUASI-FREE STATES 167 q.e.d. / Exercise. Lett] be a Bogoliubov transformation on a bosonic Fock space «F(J^). Prove that for every k £ N, there exists a constant C = C(k, U) such that we have the operator inequality on Fock space V*(M +l)feU 0, a*G = JolqJ and Ai^q < oo. Now we are able to prove the full one-to-one correspondence between quasi-free states and its generalized one-body density matrices. Theorem. Consider a bounded linear operator on Jrf? © Jrff* F:=f7 a )>0 \a* I + J7J7 with a* = JaJ and Tr7 < c». Then there exists a unique (mixed) quasi-free state G on the bosonic Fock space J-'(Jf?) such that T = Tq, the generalized one-body density matrix of G. Proof. Step 1. We will apply the previous theorem to 2 \ a* l + JjJ*j We have JAJ = A. Moreover, recall that the condition T > 0 is equivalent to 7 > J*a*(l + 7)_1aJ on J^f 168 CHAPTER 6. BOGOLIUBOV THEORY which in particular implies that Tr(aa*) < (1 + ||7||op) Tr(7) < oo. Also, A > 0 since (7 a \ / 7 + 1 a \ ' > 0, T + 5 = ' = JYJ > 0. a* 1 + J7J*/ \ a* JjJ*J By a refined analysis, we can show that A > Eq > 0. Exercise. Prove that there exists a constant e0 > 0 such that A > Sq > 0. Hint: Kei(A) = Ker(T) n Ker(r + S) = {0} and A - \ is Hilbert-Schmidt. Thus we can diagonalize A by a block operator V G namely V*AV = £ 0 0 J£J* with a self-adjoint operator £ > 0 on Jif and V = [ U V ) , V*SV = VSV* = S, Tr(VV*) < oo. JVJ JUJ* Step 2. We have 1 1 / £' 0 VTV = V*^V - -V* 0, we find that V*rV > 0, and hence f > 0. Let us show that £' is trace class. In principle, we can compute £' directly by expanding V*rV. However, here we represent another proof which is more useful later. We observe that r«s(r + «s) 7 a \ 1 0 \ a a* 1 + J7J* / \ 0 -1 j \ a* JjJ* 7(7 + 1) — aa* ja — aJjJ* \ a*j - JjJ*a* a*a - J7(7 + 1) J* / 6.5. CHARACTERIZATION OF QUASI-FREE STATES 169 is a trace class operator on Jif © Jif*. Combining with v*rv=|? 0 \ v*(r + s)v = v*rv + s = f 1 + ^ 0 ) \o i + Ji'r J V 0 J^'J* J we find that V*TS(T + S)V = VT(V«SV*)(r + S)V = (VTV)«S(V*(r + S)V) = ( e o \ /1 o \ / i+e o \ \ o i + Ji'r J \ o -l J \ o Ji'r J V o -Ji'{i + i'),r ) is a trace class operator on J^f © J^"*. Consequently, £'(1 + £') is a trace class operator on //. and hence £' is a trace class operator on Jtf. Step 3. Let us show that there exists a (mixed) quasi-free state G' on J7(Jf?) whose generalized one-body density matrix is rG, = ( ^ 0 ) = v*rv. \ o i + Ji'r j Simple case. Let us consider the case £' > 0 for simplicity Then we can define h := log (l + {i')-1) by Spectral Theorem, namely if £' = Y2 ^n\un){un\, {un} an orthonormal basis, Ai > A2 > ... > 0, then Note that /i = ^log(l + A-1)K)( un\. Tre"h = Tr i + (O = Tr r ? i Li+eJ < Tr£' < oo. 170 CHAPTER 6. BOGOLIUBOV THEORY Thus we can simply take G' the Gaussian state G' := Z^e-dT^ = Zü1 exp [- Y loS (l + A^1 Ur. Zn = Tre-dr(ft). Recall from the computation for Gaussian states, we know that ag/ = 0 and = 1 = 1 = 1 = lG' eh - 1 eioga+K')"1) - 1 ^ + _1 * ' Thus the generalized one-body density matrix of the Gaussian state G' is exactly Tl=(? ° ^| ° 1 o i + jfj* General case. It remains to consider the general case £' > 0. Again we write £' = Ara|-ura)(-ura|, an orthonormal basis, Ai > A2 > ... > 0. n For any m > 2, define tin- Then £^ > 0 is a trace class operator on Jff. The corresponding Gaussian state G'M := Z^exp [ - dr(log (l + t^)"1))" = Z"1 exp [ - Y loS (l + (Xn + M-™)-1) a*( un a[un has the one-body density matrix ^Ig'm = £m- Then we can check that G' := lim G'M exists in trace class and it is a quasi-free state with (e o \ Tq' = hm Tq' = M->oo I 0 1 + Jf J* / 6.5. CHARACTERIZATION OF QUASI-FREE STATES 171 Exercise. Consider the Gaussian states G'M as above. 1. Prove that the partition function Zm converges to a limit Zq £ (0, oo) as M —> oo. 2. Prove that G'M —> G' strongly in trace class. 3. Prove that G' is a quasi-free state and G^Zo^expf-^log^ + A"1)^! un)a{un) n0 where I = {n : A„ > 0} and Hq is the orthogonal projection onto Ker y ^2n^j a* I Hint: You can use Monotone Convergence. un a\un/ Step 4. We have constructed a quasi-free state G' such that Yq1 = VTV. Now we construct a quasi-free state G such that Tq = T. Since V £ @, there exists a Bogoliubov transformation U such that V*A(F)V = A(VF), VF £ J^®J^*. Here recall that A(f®Jg) = a(f) + a*(g), Vf,g£J?. We choose G := V*G'V. Since G' is a quasi-free state and Bogoliubov transformations leave invariant the set of quasi-free states, G is also a quasi-free state. It remains to show that Tq = T. We have the following general fact. Exercise. Let G' be an arbitrary mixed state on the bosonic Fock space «F(J^) with Tr(A/*G') < oo. Let U be a Bogoliubov transformation and V the corresponding block operator on ,3^ © // . namely V*A(F)V = A(VF), yF£j^®J^*. Prove that V*rGV = TG, with G = U*G'U. Hint: You can use Tr A*{F1)A{F2)G ={F2,TGF1), VFUF2 e Jf? ® Jff*. 172 CHAPTER 6. BOGOLIUBOV THEORY Thus we deduce that, with G = V*GV V*VGV = rG, = p ° , ) = VTV. \0 1 + J£'J* J This implies that TG = T since V is invertible. q.e.d. Finally, we turn to Theorem (Pure quasi-free states). Any pure state G = on the bosonic Fock space J7(J$?) is a quasi-free state if and only if \I> = Vfl with a Bogolliubov transformation U. Moreover, any bounded linear operator on ,3^ © .3^* r:=(7 a |>0, a* = JaJ, Tr7 0}, and n0 the orthogonal projection onto Ker(^n^/ a*(un)a(unj). In particular, if g = |XI/)(XI/| is a pure state, then v*gv is also a pure state. In this case v*gv = (v*gv)2 = Z0-2exp [~Y2{1 + A™1)°*( un)a{un) nn. 6.5. CHARACTERIZATION OF QUASI-FREE STATES 173 If I 7^ 0, then clearly Z^exp [ - Y (l + A^1)a*(ii„)a(ii„) , z^exp |-)^2(l + A~1)q*(h71)q(^ are two different Gaussian states on the sub-Fock space J7(Span(-ura : n G /)) and hence we get a contradiction. Thus if g = |XI/)(XI/| is a pure quasi-free state, then we must have 7 = 0. Thus v*gv = Hq with IIo the orthogonal projection onto f)n<£j Ker(a* (un)a(unj) = Ker(AT), namely u*|#)<#| = u*gu = n0 = Equivalently, |#)<#| = U|fi)(fi|U* = |Ufi)(Ufi| which means that \I> is equal to VCl, up to a phase factor. Step 2. Next, let us consider the generalized one-body density matrix. Recall from the proof of the previous theorem, any bounded linear operator on ffl © Jrff* T := ( 7 a |>0, a* = JaJ, Tr7 0 is a trace class operator on J^. Recall that as we argued before, the latter formula of V*rV implies that vr5(r + 5)v=(«1 + «'» 0 ) V 0 -J(,'(i + z)J" 174 CHAPTER 6. BOGOLIUBOV THEORY In particular, the fact that G is a pure quasi-free state is equivalent to G' = which is equivalent to £' = 0, and also equivalent to Tl with a self-adjoint operator h : Jif —> Jif and a linear operator k : Jif* —> Jif satisfying k* = JkJ. Here {un} is an orthonormal basis for Jif. \ Theorem (Diagonalization of quadratic Hamiltonians). Let h be self-adjoint on Jif, k : Jif* —> Jif be linear such that k* = JkJ and Tiikk*) < 00. Moreover, (h k \ > £0 > 0 on © Jf*. k* JhJ* J Then the followings hold true for the quadratic Hamiltonian hi associated to h and k: • hi is well-defined on the core domain M U (0w,n) cFc*n. M>0 n=0 6.6. DIAGONALIZATION OF QUADRATIC HAMILTONIANS 175 Moreover, HI is bounded from below HI > —Trik^h^k) and can be extended to be a self-adjoint operator on T(ffl) by Friedrichs' method. • There exists a Bogoliubov transformation U on «F(J^) and a self-adjoint operator £ > 0 on ffl such that U*HIU = dr(f) + inf(j(HI). • The unique ground state of HI (up to a phase factor) is the pure quasi-free state Vfl and inf a(M) = Tr(/i7im) + MTr (k*am). Proof. Step 1. First we prove that the quadratic form of HI can be represented in terms of the generalized one-body density matrices. In fact, for any "reasonable" state G J7(J$?) we have (^Eltf) = Tr(/i7$) + UTr(k*a*). Recall the definition of one-body density matrices {9,7Gf)* = {*,a*(f)a(g)V), {g,a*Jf) = {*, a(f)a(g)V). We have, at least formally, {V,dr(h)V) = {um,hun){^,a*(um)a(un)^) m,n>l = ^ (^m, hun)(un,^um) = ^ (un,^ y^{um, hun)u m,n>l n>l m = ^ \Un' 7*^n) = Tr(7$/i) = Tr(/i7$) (= Ti{j]J2h^2)) n>l and ^' (\ Y (^m' kJun)a*{um)a*{un) + (um, kJun)a(um)a(un)^j ^ m,n>l = K ^] (um, kJun){^>, a(um)a(un)^>) = 3ft ^ {um,kJun){um,a^Jun) m,n>l m,n>l 176 CHAPTER 6. BOGOLIUBOV THEORY = 3ft ^ 2_^(Um> kJun)um, a^Junj = 3ft(kJun, a^Junj n>l m>l n>l = 3ft^2 (^Jun, k*a^Jun^j = Tr(/c*a$). n>l The above calculation can be made rigorous for example if \I> belongs to the core domain m Q:= U (©^r™) M>0 n=0 Indeed, if Tr(A/*G) < oo, then Tr(7$) < oo and Tr (a$a^) < oo, and hence Tr(k*a^) is finite, while Tr(/i7$) is well-defined (can be +oo, but always > — oo). Step 2. Now we prove that EI is bounded from below. Recall that from > 0 we have 7$ > a$J(l +7$)~1J*a4 By the same reasoning, from A > 0 we find that /i > kJh'Krk* = rk*hrxkJ. Using the cyclicity of the trace and the Cauchy-Schwarz inequality we can estimate | Tr(ifc*a*)| = | Tr((l + 7*)1/2J*k*h~^2 ■ h^a^l + 7*)"1/2)| < ||(1 +7*)1/VW-1/2||hs • ||/i1/2«*J(l +7*)-1/2||hs. Since ||(1 + 7*)1/2^*/i-1/2||HS = ]JTt ((1 + 7*)i/2J*k*h^kJ(l + 7*)V: Tr^h^k) + Tr ^^h^kJ-f^j < JTi^h^k) + Tr (fry*) and ||/i1/2a* J(l + 7*)"1/2||hs = WTr ^V2a$ J(i + 7$)-i J*a* fci/2) < ^Tr (fry*) 6.6. DIAGONALIZATION OF QUADRATIC HAMILTONIANS 177 we find that Tr(k*a*)\ < yjTrihW-yyh1/2) ■ yjTr(kh^k*) + Tr^^i/a Here in the last estimate we have used the elementary inequality y/x{x + y) = + y/2)2 ~ y2/4 < x + y/2, Vx, y G [0, oo). Combining with Step 1, we conclude that for any $ £ Q, 1 (#,EI#) = Tr(/i7$) + KTr(£;*c^) > Tr^/i"1*;*). Thus EI is bounded from below and it can be extended to be a self-adjoint operator by Friedrichs' method. The extension, still denoted by EI, satisfies EI > —^ Tr(/c/i_1£;*). Step 3. Finally we prove that EI can be diagonalized by a Bogoliubov transformation. Finite dimensional case. To make the argument transparent, let us first consider the case when ffl is finite dimensional. Using the calculation in Step 1, we can connect the quadratic Hamiltonian EI on J7(J$?) and the block operator A on J^f © J^f* as follows: (#,EI#) = Tr(/i7$) + $tTr(k*ay) = - Tr h k \ I 7$ k* JhJ* j \a% J7*J* = l-TT{AT^)-l-TTh. By the assumptions on A, we know that there exists a block operator V G which diago-nalizes A, namely o JiJ* J for some self-adjoint operator ^ > 0 on J$?. Now let U be the corresponding Bogoliubov transformation, namely VAV* = V*A{F)V = A{VF), VFG^r©^r*. 178 CHAPTER 6. BOGOLIUBOV THEORY Then recall from a previous exercise that Thus for any pure state \I> G J7(J$?) we have (^,U*HU^) = (TO, HOT) = -Tr(^TOT) Tr h = i Tr(^VT*V) -\Tlh = \ Tr(V^VT^) -^Trh = iTr 2 2 2 0 J£J* j \al 1 + J7$ J* - Tr h 2 = Tr(^) + iTr(0-iTr(/*) = (^,dr(0^) + ^Tr(^-/t). This means that U diagonalizes HI, namely U*XW = dr(Z) + ±Tr(Z-h). Note that dr(£) > 0, with 0 is the lowest eigenvalue with the unique eigenvector. Therefore, EI has the unique ground state Vfl, with the ground state energy inf a(M) = \ Tr(^ — h). General case. The proof in the general case follows a similar strategy, except that we cannot write Tr(£) — Tr(/i) since £ and h can be not trace class separately. As in above, let V G ^ be the block operator diagonalizing A: VAV* = £ 0 0 J£J* for some self-adjoint operator £ > 0 on Jif. To proceed in the infinite dimensional case, we need Lemma. IfV£& and VAV* is block diagonal, then V* 10 o \ ( X Y )v = V o i I \ Y* 1 - \-JXJ* 6.6. DIAGONALIZATION OF QUADRATIC HAMILTONIANS 179 Let U be the corresponding Bogoliubov transformation, namely UM(F)U = A(VF), VF e jf © Jf. Then for any state e T(Jif) we can write rw = v*r*v = W 7* a* v = W 7* a* v V al 1 + J^J* ) \a% J7*J* / F* 1 + JXJ* Consequently, 7u* «w 7* Therefore, combining with the computation in Step 1, we have (^,U*IM) = (TO, HOT) = Tr(/i7OT) + KTrO*aOT) = - Tr X Y Y* JXJ* A 7u* au* au* ^ 7u* J* 1 Tr 2 AV* \ i / r. + - Tr 2 7ip Ctip a% J^J* £ 0 \ / 7* a* 0 J£J* J \ a% J7*J* Tr(^) + Ti(hX) + &Tr(k*Y) dr(O^) + Tr(hX) + 3ftTr(/c*y) Tr X Y Y* JXJ* j h k k* JhJ* X Y Y* JXJ* Thus ITHIU = dT(0 + Tr(hX) + $Tr(k*Y). Here note that 7un = X and aVn = Y. Thus we obtain the desired conclusion. It remains to prove the above technical lemma. q.e.d. Proof of the lemma. Let U : j? ->■ j? and V : * ->■ be the block components of V. Then we can write / 0 0 \ / [/* J*V*J* v = \oi/ V v* JU*J* 0 0 0 1 U V JVJ JUJ* X Y Y* 1 + JXJ* 180 CHAPTER 6. BOGOLIUBOV THEORY where X = J*V*VJ > 0, Y = J*V*UJ*. Thanks to Shale's condition Ti(VV*) < oo we obtain immediately Tr(X) < oo, Ti(YY*) < oo. Now we prove that Tr(X72/iX72) < oo, using the additional information that VL4.V* is block diagonal. By a straightforward computation of the off-diagonal term of VAV = ( U V )( h k ) (U" J"V"J" ) \ ,1V,1 JUJ* J \k* JhJ* J \ V* JU*J* J we find that Uh,J*V* + UkJU* + Vk*J*V* + VJhU* = 0. Recall from the proof of the existence of Bogoliubov transformations, we can find orthonormal bases {"Ui}i>i, {ft}i>i for J^f and \ > 0 such that U*Ui = cosh(Al)/l, J*V*Ui = sinh(Ai)/i, V* > 1. (If we change V ^ V*, then (U, VJ) \-> (U*, J*V*)). Consequently, 0 = {Ui, (UhJ*V* + UkJU* + Vk*J*V* + VJhU*)u%) = 2cosh(Al) sinh(Ai)(/i, hf%) + cosh(Al)2(/l, kJft) + sinh(Al)2(/l, J*k*f%) and hence 2cosh(Al)sinh(Al) 1. On the other hand, since {u{\ are eigenfunctions of UU*, {f{\ are eigenfunctions of U*U: j j* j j f U*UU*fi U'uiWU'mW2 2 2 UUf,= =— — = fzW U.W =cosh(Al)/l, V*>1. \\U Ui\\ \\U*ul\\ Combining with U*U = 1 + J*V*VJ = 1 + X we find that Xf% = (cosh^)2 - l)f% = sinh(Al)2/l, V* > 1. 6.6. DIAGONALIZATION OF QUADRATIC HAMILTONIANS 181 Consequently, from the above computations and the Cauchy-Schwarz inequality we have i>l i>l E (cosh(At)2 + smh(Al)2)(/l, kJfi) i> ^ 2 cosh(Aj < ^Esinn(;v»)2suP(1 + 2sinh(AJy i>l 3 t -i 1/2 r -i < Tr(FF*)(l + 2Tr(Vl/*))2 Tr(kk*)\ \2\2 i>l EiifcJ^ 1/2 i>l 1/2 r^_______2i 1/2 i>l 1/2 < oo. Thus X1l2hX1l2 is trace class. This completes the proof of the lemma. q.e.d. Chapter 7 Validity of Bogoliubov approximation In this chapter we will rigorously justify Bogoliubov's approximation for weakly interacting Bose gases. We focus on the mean-field regime where the system contains N identical bosons in Rd, described by the Hamilttonian acting on L2(Rd)®sN. Let us think of the simple situation with • Trapping potential: V £ L^c(Rd,R), lim^^ V(x) = +00; • Positive-type bounded interaction: 0 < w £ L1(lRd). (More general conditions will be discussed later.) Then we know that there exists a unique Hartree minimizer u$ > 0 and there is the complete Bose-Einstein condensation, namely • The ground state energy of HN is given by the Hartree energy to the leading order i=i lo) =N + 0(1). 182 7.1. BOGOLIUBOV HAMILTONIAN 183 In this chapter, we will prove that EN = Neu + eBog + o(l) where eBog is the ground state energy of a quadratic Hamiltonian HBog on Fock space which is predicted by Bogoliubov's approximation. More generally, we will show that the n-th eigenvalue of HN is lin{HN) = Neu + fin(MBog) + o(l), Vn = 1, 2,... We also obtain the convergence of the eigenstates for HN in terms of the Hartree minimizer uq (the condensate) and the eigenstates of HBog (excited particles). These results were first proved by Seiringer (2010) for the homogeneous gas and by Grech-Seiringer (2013) for trapped gases (the setting we consider here). We will follow the approach by Lewin-N-Serfaty-Solovej (CPAM 2015), with some simplifications. 7.1 Bogoliubov Hamiltonian Bogoliubov's theory suggests that the excited particles (particles outside of the condensation) are described by the quadratic Hamiltonian HW = ^2 {um,(h +K)un)a*man + ^ ^ {{um, KJun)a*ma*n + h.c^j m,n>l m,n>l where • {^JWo is an orthonormal basis for Jif = L2(lRd); given uq > 0 we can take all -u„'s of real-valued functions; • h is the mean-field operator associated to the Hartree equation h = —A + V + |"Uo|2 * w — ii, huQ = 0; Recall that h > 0 and uq is the unique ground state for h on ,3^. Moreover, the condition Vix) —> +oo as |x| —> oo ensures that h has compact resolvent. In particular, we have the spectral gap h > £0 > 0 on Jf+ := {uq^. 184 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION • K : Ji? —> J/f is a linear operator with kernel K(x, y) = u0(x)u0(y)w(x - y). The condition w > 0 implies that K is a positive operator on Ji? (why?). Note that we will think of HIBog as an operator on the excited Fock space = C©^T+©^2©..., jf?+ = {u0}± = QJf?, Q = l-\u0){u0\. Theorem. The Bogoliubov Hamiltonian HIBog on the excited Fock space «F(J#+) is a self-adjoint operator with the same quadratic form domain of dT(h)\jr^+y Moreover, • There exist a Bogoliubov transformation U on iF{,3^+) and a self-adjoint operator £ > 0 on //. with compact resolvent such that U*HBogU = ar(0 + eBog. • The ground state energy is finite eBog := inf cr(HlBog) G (-oo, 0]. Moreover, HIBog has a unique ground state Vfl (up to a complex phase). This ground state is a pure quasi-free state on «F(J#+). • HBog has compact resolvent and its spectrum is oo o-(HBog) = |eBog + Y2ni(il I e% G a(0,ni = 0,1, 2, ...|. i=i Remark: The ground state energy eBog is always negative (< 0) except the non-interacting case (w = 0). Proof. First, let us rewrite the Bogoliubov Hamiltonian in a form compatible to the previous chapter. It is convenient to restrict the relevant operators h, K to the subspace Since huQ = 0, h leaves invariant and we will still denote by h the restriction to Recall that m(a(h\^+) > 0, 7.1. BOGOLIUBOV HAMILTONIAN 185 Moreover, using Q = 1 — \uq)(uq\ we define K\ : ■ '//. —>■ ■ '//. and : //. —>■ //. by ÜTi := QKQ, K2 = QKJQJ*. llBog (um,(h +K1)un)a*man + ^ ^ {{umi K2Jun)a*ma*n +h.c. m,n>l m,n>l Then K* = Kx, K% = JK2J* and both Ki,K2 are Hilbert-Schmidt operators. Using the positivity of -i;, we can deduce that > 0, and hence h + K\ > 0. Moreover, we have the positivity of the block operator on © Exercise. Prove the operator inequality on Jrf?+ © J^! h + K± K2 \ A:=[ > miaih) > 0. V K2* J(/i + Kx)J* / Thus by the results in the previous chapter, we can find a block operator V G ^ and a self-adjoint operator £ on .//. such that VAV* = £ 0 0 J£J* Moreover, the corresponding Bogoliubov transformation U diagonalizes M^og on «F(J#+): U*MU = dr(0 + eBog. In particular, Heog can be defined as a self-adjoint operator on «F(J#+) and it is bounded from below: eBog = infa(HlBog) > — Ti^K^h + K^1 K2) > -00. Since h has compact resolvent, £ also has compact resolvent. So it has eigenvalues 0 < ex < e2 < ... and lim^^ en = +00. The spectrum of HBog is o-(HBog) = eBog + 0-(dT(£)) = |eBog + Yn*e* I n* = °'1'2' •"}• q.e.d. 186 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION 7.2 Unitary implementing c-number substitution Heuristically, the Bogoliubov approximation can be interpreted as HN - Men « HBog. However, this formulation is a bit formal since the operators Hn — E^ and HB0g live in different Hilbert space. This incompatibility comes from the c-number substitution which replace ao, ag (which does not preserve the particle number) by y/No (which preserves the particle number). To resolve this problem, we an operator Un from the N-body Hilbert space Jt?®aN to the excited Fock space fF(Jt?+). We start with a useful observation = t(p.y? © q.y?) ^ t(p,y?) © t{qj?). with P = \uq){uq\, q = 1 — P- Consequently, for any wave function \I> G Jt?®sN, we can write uniquely as ^ := V?0«0 + U0 ;©s^l+li0 ®s<^H-----h<£jV where ipk G J/f+ 3. To be precise, we have Definition. Let uq be a normalized vector in a Hilbert space J^. Let J^+ = {uq}1- C J/f and ao = a(uo). We define the operator Un = Un(uq), by uN: -)■ j^N{jr+) = c © j?+ © j^'2 © • • • © jef>N N N-j (-\ 7.2. UNITARY IMPLEMENTING C-NUMBER SUBSTITUTION 187 Theorem. The operator Un '■ 3l?®aN —> J7-N(Jif+) is a unitary operator with Moreover, we have the operator identities on J7-N(Jif+) for all m,n/0 UNa*0a0U*N = N - M+, UNa*manU*N = a*man, UNa*0anU*N = y/N - A/"+ an, UNa*na0U*N = a*n y/N - N+ where an = a(un) and um, un G //. when m,n/0. Remarks: The number operator A/+ on «F(J#+) = J7(QJif+) is equal to A/j^(^r+), the restriction of the number operator M = dT(l) on J7(J$?) to the subspace «F(J#+) C J7(J$?). We have the operator identities on J7(J$?): M+ = dr(Q) = M - a*(u0)a(u0). For any wave function ^at G Jif®aN, we have {u0,^]Nu0) = {^N,a*(u0)a(u0)^N) = N - {^N,N+^N). Therefore, the Bose-Einstein condensation (mq, 7^1*0) = N + o(N) is equivalent to {^N,M+^N) < N. Roughly speaking, the transformation Un(-)U^ replaces a(u0),a*(u0) by \/N — N+. Thanks to the Bose-Einstein condensation, we can think of the operator \/N — N+ as the scalar number \f~N. Thus the unitary operator provides a rigorous way to formulate the c-number substitution in Bogoliubov's argument. Recall that the Weyl operator W := W(y/Nu0) = exp (yN(a*0 - a0/ 188 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION satisfies W*a0W = ao + ^/N, W*a*0W = a*0 + y/N, W*anW = an, n^O. Thus Un looks similar to the Weyl operator. However, while the Weyl operator is defined on the full Fock space «F(J^), the operator is more appropriate to work on the N-particle space Ji?®aN. Unlike the Weyl operator, we can only write U^a*(f)a(g)U^ but it suffices for applications {UNa(f)Ulf makes no sense). • By definition U*N : '{J0+) ->■ 3?®aN ■ However, we can extend U*N to the full excited Fock space J7-N(Jtf+) by setting 0 outside J7-N(Jtf+). This extension makes Un a partial isometry from ,J^®aN to F^N(.J^+). Proof of the theorem. Part I. We will prove that n U*N : T Y//. ) ->■ je®aN, / A(0r!) 3=0 is a unitary operator with the inverse equal to Un. Step 1. We prove that U*N is a surjection, namely U*N(.34?+) = J>F®aN. Let {un}™=0 be an orthonormal basis for Jif and denote an = a(un). Recall that «F(J^) has an orthonormal basis (n0\n1\...)-1/2(a*)no(a*1)ni..n, n, = 0,1,2,... In particular, Ji?®aN has an orthonormal basis oo (n0\n1\...)-1/2(a*)no(a*1)ni..n, n, = 0, 1, 2, J^n, = N. i=0 By the definition, /A.7" A;.//. ) contains all these basis vectors, so /A.7" A;.//. ) = 3tf®aN. Step 2. We prove that \\UN$\\ Since -Vj. 7.2. UNITARY IMPLEMENTING C-NUMBER SUBSTITUTION 189 Moreover, note that dQipj = 0, and hence a™{a*Q)nyj = 0, if m > n. Thus the vectors {(ao)W_JVj }jlo are orthogonal. Moreover, iKos^-ii^^casr-v^cao^wrv,-) = {(a*r-1^,(l + a*a0)(a*r-1^) = {(a*r-1^,n(a*r-1^) = n\\(a*or-1^\\2 = ... = nl\\^\\2. Consequently, AT ii*ii2 = E 3=0 N Eini2- 3=0 Thus \\U*N$\\ = ||$|| for any $ G J^N{Jt?+). Thus is a unitary operator from to ^r^w Step 3. We prove that the inverse of is exactly equal to Un, namely if AT * := UZ* = E ^3 then We have ,N-i ,N-i AT = #) = (pu Vz = 0,1,2,JV. a0 (aQ) 3 N — j, then l(a,Q)N 3Lp3 = 0 because aoifj = 0. If i > j, then N-i < N-j, then a^Xa^"^ is proportional to (ag)*"^ £ ^®sl, but Q®^)*"^ = 0 because Qu0 = 0. Thus N-i igii / "-0 N—if *\N—i —a0 (a0) & = ■ T^N{3^) defined before is the inverse of U*N. 190 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Part II. Now we consider the action of un(-)un. Let $ = (n) u*N = a/; n>l which is equivalent to uNa*0a0un = un(n - Af+)un = n - Af+. The remaining identity is left as an exercise. q.e.d. 7.3. TRANSFORMED OPERATOR 191 f Exercise. Prove that for any f 6 Jrf?+ we have the operator identity on J7-n(j^f+) UNa*a(f)U*N = \/N — Af+a(f). 7.3 Transformed operator Given the operator Un, we can replace the heuristic approximation HN - Neu ~ Heeg- by a better one UN{HN - Neji)U*N « MBog with two operators living in the same Hilbert space «F(J#+). Let {un}^=0 be an orthonormal basis for Jrf? and denote an = a(un). We have the second quantization form Hn = Y^ Tmna*man + ^ Wmnpqa*ma*napaq 2(N m,n>0 m,n,p,q>0 where Tm„ = (iim, (-A + Wmjipg = J J um(x)un(y)w(x - y)up(x)uq(y)dxdy. Then from the above action of Un, it is straightforward to compute Un(Hn — Ne-n)UN Lemma (Transformed Hamiltonian). > We have the operator identity on the truncated Fock space 4 Un(Hn -NeR)U*N = Y,Ai 3=0 where A) - 2 Woooo n — i ' 192 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Ai = Y (T°n + WooonN X) VN ~ M+an + h.c, n>l ^ ' A2 = E {um, (h + K)un) a*man + E (um, (K|2 * w + K) un) a*maTi 1 m,n>l m,n>l iV- 1 2 [("n.^V,,.-^FE_~7_-+ h.c.j + /i.e. , m,n>l N-jV+ + h.c, m,n,p>l ^4 4 2fiV — 1) 7 rnnpqu>manaPaq- This looks complicated, but if we formally take N —> oo, then we see immediately that all Aq, Ai, As, a4 are small (o(l)), while A2 converges to the Bogoliubov transformation Heog-This will be justified rigorously later. Proof of the theorem. The computation is tedious but straightforward, using the second quantization form and the action of Un in the previous theorem. For the kinetic terms: • TooUNaQaoUN = Tqq(N — A/+). The constant TqqN is part of the Hartree energy Ne-R. Recall that gh = (u0, (-A + V)u0) + i J J \u0(x)\2w(x - y)\u0(y)\2dxdy = T00 + ^W0ooo- The other part — T00A/+ contributes to —pN+ in the first term the first term dT{h) of A2. Recall that h = —A + V + \uo\2 * w — p with V = (uo, ("A + V)u0) +11 \u0(x)\2w(x - y)\u0(y)\2dxdy = T00 + W{ 0000- TonUN^QCLnU^ = T^y/'N — Af+an with n > 1. This term and its adjoint are part of A\ TmnUNamanU'N = Tmna*man with m,n > 1. This contributes to dT(h) of A2. 7.3. TRANSFORMED OPERATOR 193 For the interaction terms we have to use the CCR to rearrange creation/annihilation operators before applying the action of Un (as we can only make sense for U'Na*(f)a(g)UN. Note that we always have the factor (2(N — l))_1W^mJipg. • UNaQaQa0a0UN = £/Araoao(aoao - ^)UN = {N — N+){N - 1 - N+). The constant 1 N W0000N{N - 1) = —W0000 2(N-1) uuuu v ' 2 is part of the Hartree energy Ne^. The term 2(jV1_1)^oooo(-2iV - 2)N+ = -W0000N+ contributes to —pM+ in the first term dY(h) of A2. The rest is A0. UNa*0a*0a0anUN = UNa*0(a0a*0 - l)anU*N = (N - 7V+)^N - N+an - ^N - N+an = (N — A/+ — l)y/N — N+an with n > 1. Combining with the same contribution from UN(ioCLoCLnO'oUN, we obtain 2 X -v-l) S Wooon{N l)y/N-M+an. 2(N n>l This term and its adjoints are part of A\. • UNaQa^aoanU^ = UNdQaoa^anU^ = (N — N+)a*man with m, n > 1. Combining with the same contribution from UNa*maQanaoUN we obtain 2x —-—— \ ' WnmnJN - Af+)a*a„ = \ ' (um, (liXnM * w)u„) ^. ^+a*a m""n 2{N~-~\) ^ W0m0n(N -M+)a*man = ^ (um, (n2 * w)un)'- N _ i m,n>l m,n>l which is part of • UNa*0a*manaQUN and UNa?ma*0aoanUN are also equal to (iV — A/+)aJ„ara with m,n > 1, but they give 2X 2(A^ll S W/o«mo(iV-A/'+)(4a™ = (nm,Knra)A^_^"+Q>ra m,n>l m,n>l which is another part of • UNd*ma*na0a0UN = UN(alla0)(ala0)UN = a*m^N - J\I+a*n^N - J\I+ 194 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION = a*ma*n^N - M+^N - M+ - 1 with m,n>l. This gives ^ > m,n>l = ^{um,KJun)aman---—--. m,n>l This term and its adjoint give us the last part of A2. a*mapa*ny/N - N+ - 5n=pa*my/N - N+ = a*ma*napy/N - N+ with m,n,p> 1. Combining with the same contribution from V'Na*ma*naQapU^ we obtain ^ X 0( AT-T\ ^ v ^rnnpOamanaP This term and its adjoint give us a3. a*mapa*naq — 5n=pa*maq = a*ma*napaq with m, n,p,q > 1. We obtain 2(N mnpq^jji^ri V q which gives us a4. This completes the computation of Un{H^ — Ne^U^- q.e.d. 7.4 Operator bounds on truncated Fock space Now we compare the transformed operator Un(Hn — Ne^U^ and the Bogoliubov Hamilto-nian Eleog-/ Lemma. Let t^N := t(M+ < N). Then ±t^N{uN(HN - Neu)U*N - MBog)l^ < CiN-^Ml + iV"1). Note that Heog acts on «F(J#+), so the particle number cut-off t-N := 1(A/+ < N). is necessary to project it to the truncated Fock space J7-N(Jtf+). Putting differently the bound 7.4. OPERATOR BOUNDS ON TRUNCATED FOCK SPACE 195 in the lemma is equivalent to <$, (l/N(HN - Neu)U*N - HBog)$) < C($, (N^Afl + A_1)$), V$ G ^Ar(^T+). Proof. For simplicity of notation we will often not write the projection t-N and think of quadratic form estimates on J7-N(Jrf?+) instead. From the previous computation, we have 4 UN(HN-Neu)U*N = YAr 3=0 We will estimate term by term. Estimate Aq. We have 0 < A0 -M/qooo jv_1 < C —. Estimate A^. Using Hartree equation huQ = 0 we have 0 = (hu0, un) = (u0, (-A + V +\u0\2 *w - n)un) = T0n + W000n- Therefore, Ai = Y (T°™ + W^N~J^i J ^N ~ N+an + h-c- n>l ^ ' - E Wooonj^~[ ^N - M+an + h-c- n>l By the Cauchy-Schwarz inequality, •A/1 \,„ .r sf ±^il +^+0. Here we have used E I^OOOnl2 = E 1(^0,^™) 12 < H^HhS < n>l n>l 196 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Choosing e = N xl2 we obtain ±A1 < CN-1/2Ml Estimate A2. We have A2= ^ (um,(h + K)un) a*man+ ^ (um, (\u0\2 * w + K) un) a*ma,, m,n>l m,n>l 1-A/+ '■ N - 1 9 2^ [{um,KJun)aman---;-+ h.c. m,n>l N - 1 W+ ^ (""" (Kol2 ^ + i^)^)fl>n1Af + l(B*X + XB) m,n>l N-l 2 where m,n>l 5* := > (um,KJun)a*ma*n, X := V(JV-^+)(JV-^+-l) iV- 1 We have ± Y (u™ (N2 *W + K) un) l Moreover, the relevant operator commutes with Af+. Therefore, ± Y ("m, (K|2 * w + K) un) a*maj~„ ^ l N-l N On the other hand, by the Cauchy-Schwarz inequality ±^(B*X+ XB) <^(eB*B + e~1X2), Ve > 0. It is straightforward to see that X2 = < y/(N-Af+)(N-Af+-l) jfA A/Y-lUl -n-~i--1 = yv-ir-TJV Af-L N-l N - 1 < C(Af+ + If N2 7.4. OPERATOR BOUNDS ON TRUNCATED FOCK SPACE 197 Exercise. Let K be a Hilbert-Schmidt operator on a Hilbert space . Let {un}n>i be an orthonormal family of Jrf?. Consider B* := V" (um,KJun)a*ma*n. m,n>l Prove that B*B < ||K||2iSA/'2. Thus 2 ±^(B*X + XB) < ^(eB*B + e^X2) < CeMl + C£-lC(A/^2+1) . By choosing e = N-1 we conclude that ±l-(B*X + XB) Q < \\w\\L«,Q (g> Q. Therefore, in the second quantization form we have 11 tt711 r oo ., /., N CAT2 ±A' * m-vW+ -1) < v- 198 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Estimate A3. Finally, we consider M = N\1 yZ Wmnpoa*ma*naP^N - N++ h.c. m,n,p>l = N _ 1UN( E WmnpOa*na*napa0 + h.c)jU*N = n\1un{^ (u™ (Q ® QWQ ® P)UP ® Uq)a*ma*napaq + h.c)jU*N m,n,p,q>0 Thus up to a transformation by Un, A3 is the second quantization of the two-body operator (N - l)"1 (Q QwQ & P + Q ® QwP &Q + h.cZ By the Cauchy-Schwarz inequality, we have the two-body inequalities ± (Q ® QwQ ®P + Q® QwP ®Q + h.c}j < 2e~1Q ® Q\w\Q ®Q + eQ® P\w\Q ® P + eP ®Q\w\P ®Q, < \\w\\Loo (le^Q ®Q + eP®Q + eQ®Py Ve > 0. In the second quantization form, we obtain ± E (urn®un,(Q®QwQ®P)up®uq)a*ma*napaq + h.c. < C^Ml + e(N - Af+)Af+ Thus C / \ / M2 \ ±A3 < y j I \ (" A'" + -AA'.)U*N < C( - ; y + :A'. ). Choosing e = N~1/2 we get ±A> < CN~l/2Afl. Thus in summary, we have prove that ±(UN(HN - Neu)U*N - HBog) < CN-^Nl + CN~X as a quadratic form estimate on J7-N(yJif+). This completes the proof of the lemma, q.e.d. 7.5. IMPROVED CONDENSATION 199 7.5 Improved condensation Recall that under the condition 0 < w G L1(lRd) for any wave function \I> G J{?®sN satisfying H^ty) < Aen + C(l) we have the complete BEC on the Hartree minimizer uq: (^,A/+^) < Oil) where A/+ = dr(Q) with Q = 1 — |ito)(wo|- Since UnAI+U^ = A/+, the same bound holds if \I> replaced by [/jv^- From the previous section, to control the error of Un{Hn — Ne^U^ — Eleog, it is desirable to have an upper bound on (^n, N+^n)- This improved condensation is proved in this section. Lemma. Assume that 0 < w G L1(Md). Let \I> G J/f^aN be an eigenfunction of with an eigenvalue /in(HN) = AeH + Oil). Then (v,NVa) < o{i). Actually, the proof below can be extended to show that Af+ty) < Cfc(l) for all k > 1. However, the case k = 2 is sufficient for our application. Proof. Step 1. From the Schrodinger equation H^ty = iin^> we have 0 = [Nl{HN - fin) + (HN - pn)Afl\ V) = 2{^,Af+iHN - fjLnW+V) + [[HN,Af+],Af+]^). Here we used the formula of "double commutator" (for any operators A, X) [[A, X],X] = iAX - XA)X - XiAX - XA) = AX2 + X2A - 2XAX. From the proof of the complete BEC, we have HN - /in> HN - AeH -C> c0N+ - C for some constants cq > 0. Therefore, 2Af+iHN - EN)Af+ > 2c0Afl - CAf^ > c0A^ - C. 200 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Thus 2{V,N+(HN - EN)Xr+V) > co(V,MlV) - C. It remains to bound the double commutator [[HN, A/+], A/+]^). Step 2. We compute [HN,Af+\. We use n>l and the second quantization form Hn = *Y Tmna*man + ^ Wmnpqa^a^apaq 2(N m,n>0 m,n,p,g>0 where rm„ = (wm, (-A + Wmjipg = yy um(x)un(y)w(x - y)iip(x)iig(y)dxdy. Since [a^, a^] = a*e[a*m, ae] = -5m=ea*m, [an, a*eae] = 5n=ean. we can compute [a*man, a*eae] = [a*m, a*eae]an + a*m[an, a*eae] = (5n=e - 5m=e)a*man. Therefore, ^ Tmn[a*man,Af+] = Y, yZTmn\.a*m.ani a*eae] m,n>0 m,n>0 m,n>0 £>1 n>l Here we have used the simplification e>i e>i e>i 7.5. IMPROVED CONDENSATION 0 if m = n = 0 or m, n > 1 1 if n ^ m = 0 — 1 if m 7^ n = 0. Similarly, ama^napaq, a}at\ = [am, a}ai\aTnapaq + a*m[a*n, a*eae\apaq + a*na* [ap, a"gae\aq + a^a^Qp^q — I SP=e + <5g=/» — <5m=/» — 5n={) a*ma*napaq Thus 2(N-1) m,n,p,q>Q 2{N - 1) 1 2(JV- 1) 1 m,n,p,q>0 £>1 m,n,p,q>0 l>\ T * * mnpqQ'rrflrfl'P^'q 2(N-1) ^ m,n,jc,g>0 m,n>l m,n,p>l In summary, [i^AT,A/"+] = y^(TQna*0an - h.c). + ^ ^(W/Waoaoao«n - h-c.) n>l n>l aOamanap I m,n>l m,n,p>l Recall that by Hartree equation T0n + Wooon = (-A + V + |-u0|2 * = (u0, (h + p)un) =0, Vn > 1. 202 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Therefore, Tona*0an + —-- y^ Wooona*0a*0aoan n>l n>l = ^TTT Y W0oon(-(N - l)a*0an + a*0an(N - Af+)) n>l YZ Wooon(a*0an(M+ - 1)) = ~J^~[ Y W000n(a*0M+an) Thus N _ _ n>l n>l [HN,Af+] = —————— ^(WooonaS^+a™ ~ n>l pfZTi Y (Woomna*0a*0aman - h.c.) + N _1 Y (W(>mnP m,n>l m,n,p>l Step 3. We compute [[HN,Af+],Af+\. Note that [A - A\N+] = [A,Af+] - [A*,Af+] = [A,Af+] + h.c. Therefore, Using [[HN,Af+],Ar+] = --^Y(wooonK^n,Ar+] + h.c.) n>l + N _ 1 ^ (Woomn[a*0a*0aman,M+] + h.c.) m,n>l m,n,p>l K, A/"+] = ajV+ - A/"+a™ = (A/"+ + l)an - Af+an = an, \/n > 1, [aman,Af+] = amanM+ - M+aman = 2aman, Vm, n > 1, [a*manap,M+] = a*manapM+ - M+a*manap = a*manap, Vm, n,p>l 7.5. IMPROVED CONDENSATION 203 we obtain [[HN,Af+],Af+] = -j^-Y^iWooonaP^+an + h.c.) + N _ 1 / J (Woomnalalaman + h.c.) m,n>l ~T7 (W0mnPa*0a*manan + h.c. ]\f _l / j ^ ' ' wtintp~[j~rn.~it-'A'p m,n,p>l Step 4. Now we estimate [[HN, A/+]A/+]^r). From the above computation we have [[HN,J\f+],Af+]ty) =h + I2 + h where n>l 12 = mN~[ Woomn{*,a0'a0'amany), m,n>l 13 = mN~[ E W0mnp{^,a*0a*manap^). We bound term by term. Estimate I\. By the Cauchy-Schwarz inequality, we have lJll <2^T^|^ooon||(^,ao^+^)| n>l <^El^ooon|||/V-+ao^||K^|| N n>l < ^H^+ao^ll JY\Wooon\2 /^IMf \ n>l \ n>l < --_=(^,A^^). 204 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Here recall that Wooo-n = (^d Kun) with K(x, y) = uo(x)w(x — y)uo(y). Estimate I2. By the Cauchy-Schwarz inequality we have AN~[ E \Woomn\\{^,a*0a*0aman^)\ \h\ < /V — I m,n>l C < - iM^oomnlllaoao^llllaman^l N m,n>l < ^IKaO^II / E \W00mn\2 I E 11°™°"^! \/ m.n>l \ m.n>l < ^y/{*,a*0a*0aoao*) I , a*ma*naman^>) \ m,n>l C Here recall that Woomn = {um, KJun) with K(x,y) = u0(x)w(x — y)u0(y). Estimate 1%. Recall that from the analysis of A3 in the previous section, we have the quadratic form estimate ± N\ 1 ( E (W0mnPa*0a*manap + /i.e.) = ± ^ ( E {um®un,(P®QwQ®Q)up®uq)a*ma*napaq + h.c. +eJW+)<-<^;. (Here we took e = iV"1/2.) Thus In summary, we have ±<#, [[//at, jV+],= ±(/i + J2 + /3) < C(^, (jv+ + l)2^) Step 5: Conclusion. We have 0 = 2{v,jv+(hn - EN)N+V) + [[hn,jv+],jv+]v) > cq(V, A/"^) - c - c{v, (jv+ + l)2^). 7.6. DERIVATION OF BOGOLIUBOV EXCITATION SPECTRUM 205 This implies that (^A/"3^) < C. Consequently, {V,NZV) < C. q.e.d. We can prove ($, A/"+$) < oo for eigenfunctions of the Bogoliubov Hamiltonian, using either the above strategy or the fact that HB0g can be diagonalized by a Bogoliubov transformation. More gerenaly, we have Exercise. Let $ G «F(J#+) be an eigenfunction of the Bogoliubov Hamiltonian Eleog-Prove that ($,A/^$) 1. 7.6 Derivation of Bogoliubov excitation spectrum Now we are able to rigorously justify Bogoliubov approximation. Recall that we are considering the Hamiltonian N 1 Hn = ^(-A^ + V{Xi)) + Y w{Xi-Xj) i=l ll m,n>l on the excited Fock space «F(J#+) where h = —A + V + \u0\2 * w — p, K(x, y) = Uq{x)w{x — y)uo(y). 206 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Theorem. Assume that • Trapping potential: V £ L^c(M.d,M), \im\x^00V(x) = +00/ • Positive-type bounded interaction: 0 < w £ L1(Md). Then the following statements hold true. • Convergence of eigenvalues. For any i = 1,2,..., the i-th eigenvalue of H^ satisfies lim (fJn(HN) - Neu - ^(HBog)) = 0. • Convergence of eigenstates. Let ^>N^ be an eigenfunction of H^ with the i-th eigenvalue ^(H^). Then up to a subsequence as N —> 00, we have lim UN^S = strongly in J7(yJ$f+), where $^-) is an eigenfunction o/HIBog with the i-th eigenvalue AiiQHBog). Remark: When n = 1, the ground state of HBog is unique (up to a phase), and hence, up to a correct choice of the phase, we have the convergence for the whole sequence N —> 00. More precisely, we can find a sequence of complex numbers {z^} , \z^\ = 1 such that lim znUnVW = Proof. To make the idea transparent, let us consider the ground state first, and then explain the extension for higher eigenvalues. Step 1: Ground state energy - lower bound. Let v&j^ £ ,yf®aN be a ground state for H^. By the validity of Hartree theory we know that {*%\HN*$) = /jl^Hn) = EN = Neu + Oil). Therefore, we have the (improved) condensation = {UN^\NlUN*V) = 0(1). 7.6. DERIVATION OF BOGOLIUBOV EXCITATION SPECTRUM 207 On the other hand, we have the operator bound on J7-N(J{?+) ±1^N{UN(HN - Neu)U*N - MBog)l^ < CN-^iNl + 1). Here recall that t-N = 1(A/+ < N) is the projection on the truncated Fock space J7-N(Jtf+). Taking the expectation against Un^n we find that HN^) = Neu + {UN^i\ Mb^UnVW) + O^-1'2). By the variational principle, {uN*%\mB^uN*(£)) >in(MBog) we conclude the lower bound Vi(HN) > Neu + /i!(ElBog) + Oi^N-1'2). Step 2: Ground state energy - upper bound. Let &A G «F(J#+) be the ground state for MBog. We know that $W is a quasi-free state, and in particular ($(i),A^$(1)) < C < oo. We can restrict $ to the truncated Fock space J7-N(Jtf+) without changing the energy too much. / Exercise. Let $ G be an eigenfunction for M.bog- Define Prove that 111- $11 ->■ 1 and lim <$A7,HlBogSjv) = ($,HBog$). By applying the above operator bound on Un(Hn — Ne-a)UN — EIBog for 208 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION and using the variational principle for Hn we obtain the upper bound »i(HN) < {U*N^\HNU*N$$) = Neu + (^ElBog*^) + 0(N~1'2) = Neu + ^(MBog) + 0(1)^00. Combining with the lower bound in Step 1, we conclcude the convergence of the ground state energy Hi(HN) = Ne-a + yUi(HlBog) + 0(1)^^. Step 3: Convergence of ground state. Let v&j^ be a ground state of Hn- From Step 1 and Step 2 we know that /ii(tfjv) = HN*$) = Neu + {UN*$, MBogUN*$) + o(l) and Hi(HN) = Ne-a + yUi(ElBog) + o(l)N=oo- Therefore, lim {UN*%\mB^UN*$) =yui(MBog). n—¥00 On the other hand, we know that HBog has a unique ground state G «F(J#+) (up to a phase) and there is the spectral gap MHlBog) > /"l(HlBog)- The convergence Un^n —> (up to a correct choice of the phase) thus follows from a standard variational technique. f Exercise. Let A be a self-adjoint operator on a Hilbert space with the min-max values satisfying fJ,±(A) < /12(A). In particular, A has a unique ground state uq (up to a phase). Prove that for any sequence {xn}n>i C Q(A) satisfying \\xn\\ = 1, {xn, Axn) ->■ /ii(A) we can find a sequence of complex numbers {zn}, \zn\ = 1 such that znxn —> uq strongly. Step 4: Higher eigenvalues - lower bound. Now we consider the lower bound for the 7.6. DERIVATION OF BOGOLIUBOV EXCITATION SPECTRUM 209 eigenvalue /il(Hn). By the min-max principle, we have HL(HN) = max i^,HN^) *gx,||*||=1 where X C D(HN) C 3i?®aN is a subspace spanned by the first L eigenfunctions vff of HN. Denote n := UNV®, Y = Span{$$ : * = 1, 2,L} = UNX C ^Ar(^T+). Then we have dim Y = dim X = L because Un is a unitary operator from 3f®sN to J7+-7V(J^+). Moreover, for any i = 1,2,L we have Therefore, max ($,A/"2$) = Oil). $gY,||#||=1 Using the operator bound ±1^N(UN(HN - Neu)U*N - MBog)l^ < CN~1/2(Ml + 1). we obtain max *GY,||*||=l ($, UNHNU*N) - Neu - ($, HBog$) < O^N-1'2). Consequently, by the min-max principle we conclude that MHBog)< max <$,HBog$)< max ($, UNHNU*N) - Neu + O^"1/2) $gy,h*h=i $gy,||*||=i < max HNV) - NeB + 0(A^-1/2) *gx,||*||=1 = pL(HN)-Neu + 0(N-1/2). Thus we obtain the desired lower bound l-tL{HN) > NeH + Pl(Mbos) + OiN-1'2). Step 5: Higher eigenvalues - upper bound. We use the min-max principle again. Let 210 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION ^\ ... , $(L) be the first L eigenfunctions of HIBog. Define Then by an extension of a previous exercise we know that for all i,j 6 {1, 2,L}, lim = ($«,$^) = dl=J N—>oo and lim <$S,HBog$!?) = <$(i),MBog*0)) = 5l=^(HBog). N—¥oo Consequently the space Y := Span{$g,i = 1,2,..., L} C T-N{^+) satisfies dimV = L, lim max ($, H^g^) = /^zA^Bogj $GY,||$||=1 Since for any i = 1,2,L ($J,A^$J) < (^(l),A^^(l)) = Oil), Vi = l,2,...,L we have max ($,A/"2$) = 0(1). $GY,||$||=1 Thus from the operator bound il^M/^v - NeB)lTN - MBog)l^ < CN-^iNl + 1) we obtain max $gy, 11*11=1 ($, UNHNU*N) - Nev - ($, HBog$) < 0(iV-1/2). By the min-max principle we conclude that ^ , -1/2) >^L(/^)-AeH + 0(iV-1/2) yUL(HBog) = max ($,MBog$)> max {,UNHNU^) - NeB +0(N~ $gy, 11*11=1 #eY,||*||=i 7.6. DERIVATION OF BOGOLIUBOV EXCITATION SPECTRUM 211 which is equivalent to the upper bound Pl(Hn) < iVeH +^L(MBOg)0(iV-1/2). Combining with the lower bound in Step 4, we obtain the convergence of eigenvalues pL(HN) = Neu + /iL(MBog) + O^N-1'2). Step 6: Convergence of eigenfunctions. In Step 4 we have proved that if ... , ^ffi are the first L eigenfunctions of HN, then the vectors d>« := UN*oo This implies that up to a subsequence as N —> oo, the sequence {3>J^}jv converges strongly to an eigenfunction of Heog with eigenvalue ^(HBog), thanks to the following abstract result (recall that HB0g has compact resolvent). This completes the proof of the theorem. q.e.d. f > Exercise. Let A be a self-adjoint operator on a Hilbert space. Assume that A is bounded from below and that the first L min-max values satisfy Pi < P2 < ■■■ < Pl < inf cress(A). Consider the vectors {^}^>i~i satisfying lim = 5i=j, lim{x3n,Ax3n) = p3, Vz, j G {1, 2,L). Prove that up to a subsequence as n —> oo, the sequence {x^}n converges strongly to an eigenfunction of A with eigenvalue p^. 212 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION 7.7 Extension to singular interaction potentials In this section, let us quickly explain how to adapt the previous strategy to justify Bogoli-ubov's approximation for singular interaction potentials. We consider the Hamiltonian N 1 i=l l max(d/2, 2). • The Hartree minimizer u$ > 0 is unique (up to a phase) and non-degenerate: ( h + K1 K2 \ > e0 > 0 on J^f+ © Jf??. \ K* J(h + Ki)J*)~ + Recall that h = -A + V + \u0\2 * w - fi, Kx := QKQ : .//. ->■ .//. . K2 = QKJQJ* : //. —> //. with K the operator on ffl with kernel K(x, y) = uq(x)w(x — y)uo(y). • Any minimizing sequence of the Hartree functional has a subsequence converging to u$ (up to a phase) strongly in L2(Rd). Remarks: • The first condition on the potentials ensures that Hn is a self-adjoint operator in the same domain of the non-interacting Hamiltonian, by Kato-Rellich theorem. In particular, Coulomb potential w{x) = l/\x\ with x G R3 is allowed. • The second condition means that the Hessian of the Hartree functional at the minimizer uq is non-degenerate. This ensures that the Bogoliubov Hamiltonian is well defined (see below). • The third condition ensures that we have the complete BEC: for (^n, Hn^n) = Ne-R + o(N), lim = L N^oo N 7.7. EXTENSION TO SINGULAR INTERACTION POTENTIALS 213 Recall the Bogoliubov Hamiltonian HW = Y + K)un)a*man + i ^ ((nm, KJun)a*ma*n + /i.e.) m,n>l m,n>l on the excited Fock space «F(J#+), where a„ = a(un) with {"UjjJ^Iq an orthonormal basis for = L2(Rd). -\ Theorem. Under the above assumptions, the Bogoliubov Hamiltonian HBog on the excited Fock space J7{,3^'+) is a self-adjoint operator with the same quadratic form domain of &T{h)\r(^+y Moreover, • There exist a Bogoliubov transformation U on «F(J#+) and a self-adjoint operator £ > 0 on .'//. such that iraBogu = dr(0 + ^i(EIBog)- Moreover, inf cr(£) > 0 and aess(£) = aess{h\^+). • EIBog has a unique ground state Vfl (up to a complex phase). Moreover, it has the spectral gap between the second and the first min-max values mhbog) - /ii(MBog) = inf 0. We have the operator lower bound 1 Bog > -dr(Q(-A + V+ + 1)Q) - C. Proof. Step 1. From the condition w G Lp(Rd) + L°°(Rd) and u0 G H\Rd) we find that K is a Hilbert-Schmidt operator II*& = // \Ki,,y)?My = // M,)l>(, - y)\>M)\>** < oo. J Jm.dxM.d J Jm.dxm.d To see the the latter bound we can use Sobolev embedding H1^) C Lq(Rd) for 2 < q < 2* (recall 2* = +oo if d < 2 and 2* = 2d/(d — 2) if d > 3) and Young's inequality. More 214 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION precisely, by linearity we can assume w £ Lr(Wi) with max(d/2, 2) < r < oo and estimate \u0(x)\2\w(x - y)\2\u0(y)\2dxdy < ||«o|lL(R")lklli,'-(R2, - H---h - = 1. q r q The condition r > max(d/2, 2) allows us to take q < 2* . Step 2. Since K is Hilbert-Schmidt, K\ and K2 are also Hilbert-Schmidt operators. Thanks to the non-degeneracy of the Hessian ( h + K-L K2 \ A := > e0 > 0 on Jť+ © .///. V ií2* J(h + Ki)J*)- + we can apply the diagonalization procedure discussed in the previous chapter to HW = Y + Ki)un)a*ma„ + ^ ^ ((wm, K2,lun)a*ma*n + /i.e.) m,n>l m,n>l Thus there exist a Bogoliubov transformation U on ^(J^) and a self-adjoint operator £ > 0 on such that U*MBogU = dr(£) + yu1(MBog) Here the ground state energy yUi(ElBog) is finite. Let us prove that o-essiO = <7ess(h), h:=h + Kx. In fact, from the diagonalization procedure we also know that there exists a block operator v=f u v )ew \ ,1v,1 JUJ* J (which is associated to U) such that (i 0 )=vav=( u v )( 1 KJ )(u" J"V"J") \ 0 J i J* J \ ,1v,1 JUJ* J \ K*2 JhJ* j \ V* JU*J* I 7.7. EXTENSION TO SINGULAR INTERACTION POTENTIALS 215 A direct computation shows that £ = UhU* + UK2V* + VK*2U* + VJhJ*V*. Since U is bounded and V, K2 are Hilbert-Schmidt, UK2V* and VK'2U* are trace class. Moreover, by a lemma from the proof of the diagonalization of quadratic Hamiltonians, we know that Ti(X1/2hX1/2) < oo, X := J*V*VJ. The latter implies that 1^1/2^*11^ = ^(h^rv^jl1/2) = Tiih^Xh1'2) < oo and similarly ||W/i1//2||hs < °o- Thus VJhJ*V* is trace class. In summary, we have proved that £ — UhU* is a trace class operator, therefore CeSS(£) = aess(UhU*). Since [/[/* — 1 and U*U — 1 are trace class (thanks to Shale's condition), we deduce that CeSS(£) = aess(UhU*) = aess(h). Exercise. Let B be a self-adjoint operator on a Hilbert space. Let U be a bounded operator such that U~x is bounded and UU* — 1 is a compact operator. Prove that So, we have h + K1>e0>0. Consequently, inf cress(£) = inf aess(h + Kx) > e0 > 0. Since ^ > 0, we deduce that inf 0. In fact, if infcr(£) = infcress(£), then obviously infcr(£) > Eq > 0. On the other hand, if infc(£) < infcess(£), then by the min-max principle, infc(£) is an eigenvalue of £, which must be strictly positive since ( > 0 as an operator. Step 4. Using u*MBogU = dr(0 + /ii(EiBog) and inf cr(£) > 0, we find that Heog has a unique ground state Vfl (up to a phase). Moreover, it satisfies the spectral gap mhw) " MHBog) = /i2(dr(0) - /ii(dr(0) = inf a(0 > o. Step 5. Now we prove the operator lower bound for HB0g — Ati(ElBog)- Since inf cr(£) > 0, we have U*MBogU - /ii(ElBog) = dr(0 > mi ^U*(AE)U - C. In the second inequality, we have used U*A/"+U < C(JV+ + 1). Thus we have proved that HBog > - C. Next, let us consider the Bogoliubov Hamiltonian in more detail. ElBog = E + K)un)a*man + ^ E ((Mm,^n)aX + ^-C-j m,n>l m,n>l 7.7. EXTENSION TO SINGULAR INTERACTION POTENTIALS 217 Since K is a Hilbert-Schmidt operator, we have ±\ /Z ((um,KJun)a*ma*n + h.c}j l Moreover, the condition V_, w G Lp(Rd) + L°°(IRd) with p > max(d/2, 2) ensures that \V-\ + |tj0|2 * \w\ < h-A) + C. Thus 2 h + K = -A + V + K|2 *w-n + K> i(-A + y+ + 1) - C. In the second quantization, we find that dr(/i + k) > ^dr(-A + v+ +1) - cw+- Thus we conclude that HBog > ^dr(-A + v+ + i)- c(xr+ +1) > ^dr(-A + v+ + i)- c(uBog + c) which is equivalent to HBog > ^dr(-A + V+ + T)-C. This completes the analysis of the Bogoliubov Hamiltonian. q.e.d. Theorem. Under the assumptions in the beginning of this section, the following statements hold true. • Convergence of min-max values. For any i = 1, 2,the i-th min-max value of Hn satisfies lim (fjLi(HN) - Neu - ^(MBog)) = 0. iv—»00 V / • Convergence of eigenstates. Assume that PL^-Bog) < hif <7ess(HIBog) for some L > 1. Then iiL^Bog) is an eigenvalue for HBog and for N large, /il(Hn) is an eigenvalue of Hn- Moreover, if is an eigenfunction of with eigenvalue 218 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION /il(Hn), then up to a subsequence as N —> oo, we have lim UnVW = $(L) strongly iniF(y,3^+), where is an eigenfunction o/HiBog with eigenvalue //l(HI Bog J The proof of this theorem follows the general strategy we discuss before, namely we will compare Un(Hn — Ne-a)UN with HiBog- However, since the interaction potential w may be unbounded, the analysis is more complicated in several places. Here is a quick explanation of necessary modifications: • We need to modify the operator bound on truncated Fock space. For example, we cannot use \w\ < \\w\\loo anymore, and hence we cannot simply bound A := UN(HN - Neu)U*N - HBog in terms of A/+. We have to use some kinetic part to control the error, resulting the following bound on t-m{^+) with l«M M). This is an analogue of the IMS localization formula, but now the local functions (fj(x) with x £ Rd are replaced by functions of number operator A/+. The part fuAfu can be controlled by the above operator bound, provided that M <^N. The part guAgu is bounded by the variational principle 9uAgM > Pi(A)g2M > ^(A)-^-. Thanks to the condensation ($, J\f+$) l ^ ' A2 = Y (um,{h + K)un) a*man+ ^ (um, (\u0\2 * w + K) un) a*man _ 1 -A/"+ m,n>l m,n>l + 2 ^ \{um,KJun)anam---—--+ h.c.j, m,n>l ^3 = ^ \ 1 Y WrnnpQa*manaP\f N -N+ + h.c, m,n,p>l AA = 2^ _ Wmnpqamanapaq- m,n,p,q>l We will estimate term by term. Estimate Aq. We have Ml 0 0. 220 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Restricting to the subspace J7-M(J^+) and taking e = y/M/N and , we have ± 1^%1^M < ('I ''(- ■ A'.) < C^|^M(HBog + C)l^M. Estimate A2. We have proved that ±(A2-MBog) max(d/2, 2) and Sobolev's embedding theorem we obtain the two-body inequality \w(x-y)\ l l N ~~[un{ zZ iurn®un,(Q ®QwQ ® p)up®uq)a*ma*napaq +h.cZjUN Thus up to a transformation by Un, a3 is the second quantization of the two-body operator (N - l)"1 (Q ® QwQ ®P + Q® QwP ®Q + h.cZj. From the assumption w G Lp(M>d) + L°°(lRd) with p > max(d/2, 2) and Sobolev's embedding theorem we obtain the two-body inequality ±(q® QwQ ®P + Q® QwP ®Q + h.cZj < 2e~1Q ® Q\w\Q ® Q + e(Q ® P\w\Q ®P + P® Q\w\P ® q), < Ce-1 ((q(-a + 1)q) ®Q + Q® (q(-a + 1)q)) + Ce ((q(-a + 1)q) ® P + p © (q(-a + 1)q)), Ve > 0. In the second quantization form, we obtain ±^3 < ^ + eiv)dr(q(-a + 1)q). £, where 222 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Pi = l(M = i). Let f,g : 1 f(x) = 1 for x < 1/2 and f(x] I ->■ [0,1] = 0 for x be smooth functions such that f2 + g2 = 1, > 1. For any M > 1 define /m := W/M) , gM ■= g(Xi/M). Then ±(A-fMAfM - giwAgM^ £. Combining with a similar formula for jm, we arrive at 1 oo A - fuAfu - gMAgM = ^ E [(/(W - f(j/M))2 + (g(i/M) - g{j/M)f PAPr i 0 we have the Cauchy-Schwarz inequality ±(PiAPj + h.c.) < PiAPi + PjAPj. Thus we conclude that ± (a - fMAfM - guAgu^j 1 oo = ±1 E [WIM) - fU/M))2 + (9(i/M) - g(j/M))2] (PAP, + P3AK i (1 — e) A + V)l — CeN. Now we are ready to provide Proof of the theorem. Step 1. Ground state energy - lower bound. Denote HN = UN(HN - Neu)U*N. Let us prove that Pl(HN) > /Jl(ElBog) + O(l)jv->oo- 224 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION By applying the localization formula for A = H^ — ^(Hn), with £ = 2 and 1 A*i(HlBog) > ~~ °o we find that ' M 2 m > ^i(HBog) - C\J — fM. The part guH^gu can be bounded from below by the variational principle 9uHNgM > Pi(HN)g2M = pi(HN){\ - fy). The localization error is controlled by the rough estimate [UNHNU*N]Aiag < C{UNHNU*N + CN) which implies [#Ar]diag < C(HN + CN), \fn(HN)\ < CN. In summary we have the operator inequality on J7-N(Jif+) ~ ~ ~ C ~ ~ Hn > ImHn/m + gu^n^m — ~yj2^N ~ ^1 (-^Ar)]diag > MEW " C\J^}f2m + Pi(Hn)(1 - fM) - ^~2(Hn) + CN) 7.7. EXTENSION TO SINGULAR INTERACTION POTENTIALS 225 which is equivalent to (1 + CM-2)HN > mMBog) c CN iM2' By the assumption on the condensation, we can take a wave function tyN G jffp®N such that (fjv^jvfjv) N) = o(l)N_ ($at, f2M$N) = 1 - ($at, g2M$N) = 1 + o(i; The choice N1'2 < M ensures that N/M2 (1 + CM-2){$N, HN$N) > /ii(MBog)-C = /ii(MBog)-C Using the rough information M N M ($N, f2M$N) + iniHvXl - ($N, f2M$N)) CN iM2 :i + o(v H!(HN)o(l)N- we conclude that This is equivalent to |/ii(HBog)| = 0(1), M#jv)I = C7(JV) Pl(HN) > ytii(ElBog) + O(l)jv->oo- Vi{HN) > Neu + ytii(ElBog) + 0(1)^00. 226 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Step 2. Ground state energy - upper bound. We use the localization formula for the Bogoliubov Hamiltonian C / \ Hfeog — fM^-BogfM + <7MHlBog<7M — —(jHlBog]diag + Pl(^Bog)j C Using again > ImElBog/M + <7MHlBog<7M — —p (^Bog + Cj . we have ±/m - HBog)/ii/ < Cy -7/M(HBog + C),fM /ikfElBog/M > + C\J~j~^J ImHnJm — C\J~j~-Combining with the variational principle we have the operator bound on Fock space C / \ Hfeog _ /jW"E[Bog/M + fl'MElBogfl'M ~ (^Bog + CJ > (l + C^)"VmW(^) + tfi/MlW - ^HBog - C\J^ > which can be rewritten as 'Mn-i , ,~ , , , _ , „ /M 'i + CM-2)eBog > (i + cy —) f2MPi{HN) + ^/ii(MBog) - cv ^ Now take be the ground state for ElBog- Then {$(1),Af+$(1)) < C < oo. By choosing 1 < M < N we obtain (7^$(1)) < (A/"+/M)$(1>) = o{\), = 1 + o(l). 7.7. EXTENSION TO SINGULAR INTERACTION POTENTIALS 227 Thus 1 + CM-2)/ii(ElBog) = (l + CM"2) HBog$(1)) = (1 + o(l))/i1(^JV) + 0(l)/i1(ElBog) + o(l). Using the rough estimate /ii(MBog) = Oil), ^(HN) = Oil) we conclude that /ii(HBog) >/ii(£jv) + o(l). This is equivalent to Pi(HN) < Ne-a + yUi(HlBog) + 0(1)^00. Combining with the lower bound in Step 1, we conclude the convergence of the ground state energy Hi(HN) = Ne-a + yUi(HlBog) + 0(1)^^. Step 3: Convergence of ground state. Take any wave function "$>N G Ji?®aN such that ^N,HN^N)<^{HN) + N~\ £w := i-A^M = 0(1)^. From Step 1 and Step 2, we obtain that the vector = Un^n satisfies (/M$7V,HlBog/M$7v) II/m* yUl(MBogJ. n\ Thanks to the spectral gap yUi(ElBog) < /^(HBog) we concm■ 0, ||/m*aHI ->■1 we find that Jm^n and hence &N —> strongly in Fock space F{Jti?+). 228 CHAPTER 7. VALIDITY OF BOGOLIUBOV APPROXIMATION Step 4: Convergence of min-max values and higher eigenstates. By the same analysis in Step 1 and Step 2, plus the min-max principle, we also obtain the convergence of all min-max values Vi{HN) = Neu + Hi{UBog) + 0(1)^00, Vz = 1, 2,... In particular, this implies that for N large, HN also have the spectral gap P2{HN) - Hl(HN) = yU2(ElBog) - yUi(ElBog) + 0(1)^^ > 0. Consequently, HN has a unique ground state (up to a phase). More generally, if yUi(ElBog) < ••• < /^(HlBog) < inf <7ess(HlBog), then for N large, we have Pi(HN) < ... < Hl(Hn) < mfaess(HN). Consequently, Hn has at eigenvalues /ii(Hn), /il(Hn). If v&j^ are the corresponding eigenfunctions, then the vectors 1 1 satisfy lim {*», *) = S,=1, lim HBog*g) = ft(HB„g). N—>oo N—>oo By a previous exercise, this implies that up to a subsequence as N —> oo, the vector v&j^ converges strongly to an eigenvector $W of HB0g with eigenvalue ^(HlBog)- Thanks to the condensation, we have ||0m*$||->o, ii/m^II-^i. Thus up to a subsequence as N —> oo, the vector converges strongly to an eigenvector $W of HBog with eigenvalue //j(HlBog), for all 1 < i < N. This completes the proof of the theorem. q.e.d.