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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.

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I also find the theory very beautiful, but I know very few physicists who leverage it. Of all the physicists in my department I know of only 1 group that regularly uses Fredholm theory. I'm currently using it for a project, but neither the problem nor the result are simple. –  Kevin Driscoll Apr 15 '13 at 19:48
Although not a simple example, Fredholm's Alternative Theorem is used to derive an adjoint equation to the linear stability equations used in fluid dynamics. The adjoint and direct equations can then be used to decompose a flow field into the constituent modes of the linear stability equations. –  OSE Apr 15 '13 at 20:57
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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.

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