Q3.ll Inverse scattering about — oo. Find the equation for L(x,z) if iA_(x;k) = e~"" +j    L(x,z)z~'kz dz
is a solution of    + {/. — u(x)}ip = 0; see equation (3.39). What boundary conditions must L(x, z) satisfy? Q3.12 Poles of the transmission coefficient.
(i) Use the identity (3.43) to prove that a' ' has zeros at k = \k„.
(ii) Use the identity (3.44), et seq., to show that a(k) has simple poles at
k = in„.
Q3.13 Integral equations. Find the solutions of the integral equations
(i) K(x,z)+t'(I+z) + f  K(x,y)e'(>' + 2>dy = 0;
(ii) </>(x,z) + xz+     c/)(x, y)yz dy = 0;
(iii) K(x,z)-e~
(iv) (/)(x)=1 +
K(x,>')e-(> + z|dy = 0;
c/>(y)sin(x + >')d>\
o
Q3.14 Neumann series. Now use a Neumann series to find the solution to Q3.13(i). Q3.15 Inverse scattering. Reconstruct the potential function, u(x), for which the reflection coefficient is
b(k) =-P/(P + \k), p<o.
Q3.16 Inverse scattering with zero reflection coefficient. For the case of three discrete eigenvalues, with b(k) = 0 for all k, find an expression for \A \ (see example (ii), §3.4) so that the potential function can be written as
d2
m(x)= -2-^logMI