Fredholm Theory - Inhomogeneous Equations

Inhomogeneous Equations

The inhomogenous Fredholm integral equation

may be written formally as

which has the formal solution

A solution of this form is referred to as the resolvent formalism, where the resolvent is defined as the operator

Given the collection of eigenvectors and eigenvalues of K, the resolvent may be given a concrete form as

with the solution being

A necessary and sufficient condition for such a solution to exist is one of Fredholm's theorems. The resolvent is commonly expanded in powers of, in which case it is known as the Liouville-Neumann series. In this case, the integral equation is written as

and the resolvent is written in the alternate form as