Mathematical Methods for Physics and Engineering : A Comprehensive Guide

23.7 SCHMIDT–HILBERT THEORY

Let us begin by considering the homogeneous integral equation

y=λKy,

where the integral operatorKhas an Hermitian kernel. As discussed in sec-

tion 23.3, in general this equation will have solutions only forλ=λi,wheretheλi

are the eigenvalues of the integral equation, the corresponding solutionsyibeing

the eigenfunctions of the equation.

By following similar arguments to those presented in chapter 17 for SL theory,

it may be shown that the eigenvaluesλiof an Hermitian kernel are real and

that the corresponding eigenfunctionsyibelonging to different eigenvalues are

orthogonal and form a complete set. If the eigenfunctions are suitably normalised,

we have

〈yi|yj〉=

∫b

a

y∗i(x)yj(x)dx=δij. (23.49)

If an eigenvalue is degenerate then the eigenfunctions corresponding to that

eigenvalue can be made orthogonal by the Gram–Schmidt procedure, in a similar

way to that discussed in chapter 17 in the context of SL theory.

Like SL theory, SH theory does not provide a method of obtaining the eigen-

values and eigenfunctions of any particular homogeneous integral equation with

an Hermitian kernel; for this we have to turn to the methods discussed in the

previous sections of this chapter. Rather, SH theory is concerned with the gen-

eral properties of the solutions to such equations. Where SH theory becomes

applicable, however, is in the solution of inhomogeneous integral equations with

Hermitian kernels for which the eigenvalues and eigenfunctions of the corre-

sponding homogeneous equation are already known.

Let us consider the inhomogeneous equation

y=f+λKy, (23.50)

whereK=K† and for which we know the eigenvaluesλiand normalised

eigenfunctionsyiof the corresponding homogeneous problem. The functionf

may or may not be expressible solely in terms of the eigenfunctionsyi,andto

accommodate this situation we write the unknown solutionyasy=f+

∑
iaiyi,
where theaiare expansion coefficients to be determined.

Substituting this into (23.50), we obtain

f+

∑

i

aiyi=f+λ

∑

i

aiyi

λi

+λKf, (23.51)

wherewehaveusedthefactthatyi=λiKyi. Forming the inner product of both