Physics Applications of Fredholm Theory: I find Fredholm theory beautiful, especially the Liouville-Neumann series for solving Fredholm integral equations of the second kind. There seems to be a consensus that these equations are quite useful for physicists, but all I have read is "these are useful in physics but we don't have time for that here . . . blah, blah, blah compact operators blah blah blah existence and uniqueness blah blah blah Banach spaces blah blah blah". (I love all these topics, but the point stands.) 
So my question is: Is there a simple problem in physics, perhaps suitable for advanced undergraduates, that contorts itself into a Fredholm integral equation? I reject the Poisson equation as an example, because it is too simple and there is no reason to learn Fredholm theory to solve it. 
 A: Surely. Lets consider scattering of a 1-D particle on a small potential barrier. To solve the problem, we will find energy eigenstates:
$$
H|\psi\rangle=E|\psi\rangle
$$
Set $H=p^2/2m+\epsilon U=H_0+\epsilon U$, where $\epsilon$ is a small parameter. Consider the equation
$$
(H_0-E)|\psi\rangle=|\phi\rangle
$$
Let us write a solution as
$$
|\psi\rangle=|\psi_0\rangle +G_0(E)|\phi\rangle
$$
where $|\psi_0\rangle$ lies in $E$-eigenspace of $H_0$ and $G_0(E)$ is the operator with kernel (in coordinate rep) being the casual Green function of $H_0-E$.
Now we write the original problem:
$$(H_0-E)|\psi\rangle=\epsilon U|\psi\rangle$$
so
$$
|\psi\rangle=|\psi_0\rangle +\epsilon G_0(E)U|\psi\rangle
$$
We want this to reduce to a free particle moving from left to right for $\epsilon=0$, so we write in the coordinate representation $|\psi_0\rangle=\exp(ip_Ex),\,p_E^2/2m=E,p_E>0$, and the equation:
$$
\psi(x)=\exp(ip_Ex)+\epsilon\int_{-\infty}^x G_0(E,x-x')U(x')\psi(x')dx'
$$
This is just what you want, with $K(x,y)=G_0(E,x-y)U(y),\lambda=\epsilon$, it is a Volterra second kind integral equation, and partial sums of Liouville-Neuman series give a perturbative solution to 1-D scattering.
