Love--Lieb integral equations: applications, theory, approximations, and computation

This paper is concerned mainly with the deceptively simple integral equation $$u(x) - \frac{1}{\pi}\int_{-1}^{1} \frac{\alpha\, u(y)}{\alpha^2+(x-y)^2} \, \rd y = 1, \quad -1 \leq x \leq 1,$$ where $\alpha$ is a real non-zero parameter and $u$ is the unknown function. This equation is classified as a Fredholm integral equation of the second kind with a continuous kernel. As such, it falls into a class of equations for which there is a well developed theory. The theory shows that there is exactly one continuous real solution $u$. Although this solution is not known in closed form, it can be computed numerically, using a variety of methods. All this would be a curiosity were it not for the fact that the integral equation arises in several contexts in classical and quantum physics. We review the literature on these applications, survey the main analytical and numerical tools available, and investigate methods for constructing approximate solutions. We also consider the same integral equation when the constant on the right-hand side is replaced by a given function.