Date: Nov 16, 2013 3:28 PM
Author: golabidoon@gmail.com
Subject: Eigenfunction question of a simple operator

Hello,

Consider the integral operator defined via the following kernel k(x,y):

int_{-L}^L k(x,y)*f(y) dy = x^2*f(x)-c*f''(x)

where c>0 and L>0 are some given parameters. This operator is positive definite and self adjoint. So it must have some eigen decomposition.

Can any one help with computing the eigenfunctions and corresponding eigenvalues of this operator? I know eigenfunctions v_k(x) and eigenvalues a_k must satisfy:

forall x in [-L,L]: x^2*v_k(x)-c*v_k''(x)=a_k*v_k(x)

However, I have no clue how to find the solution of this simple equation.

Thanks,

Golabi