Solving a singular integral equation for the one-dimensional Coulomb problem

Abstract

A new integral equation that describes the behavior of the momentum space wave function for the one-dimensional Coulomb potential is proposed. The obtained result turned out to be a homogeneous Fredholm integral equation of the second kind and a singular integral equation, because its kernel has a singularity at some point in the momentum space. A nontriviality of the method of solving this singular integral equation lies in the application of the integral representation for its integral kernel. The technique applied in this paper made it possible to show that the wave function in the momentum representation is simultaneously a solution of the homogeneous Fredholm integral equation of the second kind and of the linear Volterra integral equation of the second kind. Since a linear Volterra integral equation of the second kind was easily transformed into a second order linear inhomogeneous differential equation with constant coefficients, the eigenfunctions and eigenvalues in the one-dimensional Coulomb problem were found without any difficulties. Such a circumstance may indicate the validity of the new integral equation and the proposed method of its solving.