Mathematical Methods for Physics and Engineering - Matematica.NET

23.7 SCHMIDT–HILBERT THEORYLet us begin by considering the homogeneous integral equationy = λKy,where the integral operator K has an Hermitian kernel. As discussed in section23.3, in general this equation will have solutions only for λ = λ i ,wheretheλ iare the eigenvalues of the integral equation, the corresponding solutions y i beingthe eigenfunctions of the equation.By following similar arguments to those presented in chapter 17 for SL theory,it may be shown that the eigenvalues λ i of an Hermitian kernel are real andthat the corresponding eigenfunctions y i belonging to different eigenvalues areorthogonal and form a complete set. If the eigenfunctions are suitably normalised,we have〈y i |y j 〉 =∫ bay ∗ i (x)y j(x) dx = δ ij . (23.49)If an eigenvalue is degenerate then the eigenfunctions corresponding to thateigenvalue can be made orthogonal by the Gram–Schmidt procedure, in a similarway 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 eigenvaluesand eigenfunctions of any particular homogeneous integral equation withan Hermitian kernel; for this we have to turn to the methods discussed in theprevious sections of this chapter. Rather, SH theory is concerned with the generalproperties of the solutions to such equations. Where SH theory becomesapplicable, however, is in the solution of inhomogeneous integral equations withHermitian kernels for which the eigenvalues and eigenfunctions of the correspondinghomogeneous equation are already known.Let us consider the inhomogeneous equationy = f + λKy, (23.50)where K = K † and for which we know the eigenvalues λ i and normalisedeigenfunctions y i of the corresponding homogeneous problem. The function fmay or may not be expressible solely in terms of the eigenfunctions y i ,andtoaccommodate this situation we write the unknown solution y as y = f + ∑ i a iy i ,where the a i are expansion coefficients to be determined.Substituting this into (23.50), we obtainf + ∑ ia i y i = f + λ ∑ ia i y iλ i+ λKf, (23.51)wherewehaveusedthefactthaty i = λ i Ky i . Forming the inner product of both817