Novel properties of the Birman-Schwinger operator of the one-dimensional Coulomb Hamiltonian

Abstract

This paper is the continuation of a previous one devoted to the study of the self-adjoint realisation of the one-dimensional Coulomb Hamiltonian determined by a Dirichlet boundary condition at the origin. The main result here is to prove that, in this case, the Birman-Schwinger operator can be used to solve completely the eigenvalue problem for the discrete spectrum for the one-dimensional Coulomb Hamiltonian. This is possible because, as shown herein, this eigenvalue problem reduces to finding the solution of an integral equation which happens to be the integral counterpart of Laguerre’s renowned differential equation. Our findings are relevant to the treatment of the s-states of the three-dimensional Coulomb Hamiltonian due to the absence of the centrifugal term in the Hamiltonian for such states. Finally, we replace the Coulomb Hamiltonian with the one-dimensional Yukawa potential and obtain an estimation of its eigenvalues.