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

TL;DR

This paper studies the Love--Lieb integral equation, exploring its applications in physics, analyzing its theoretical properties, and reviewing numerical and approximation methods for solving it.

Contribution

It provides a comprehensive review of the Love--Lieb integral equation, including its theory, applications, and various approaches for numerical approximation and solution construction.

Findings

Unique continuous real solution exists for the equation.

Numerical methods can effectively approximate solutions.

Applications span classical and quantum physics contexts.

Abstract

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.