The one-dimensional Coulomb Hamiltonian: Properties of its Birman-Schwinger operator

TL;DR

This paper investigates the properties of the Birman-Schwinger operator for the one-dimensional Coulomb Hamiltonian, analyzing its behavior on different domains and its approximations, revealing it is Hilbert-Schmidt but not trace class.

Contribution

It provides a detailed analysis of the Birman-Schwinger operator for the 1D Coulomb Hamiltonian, including convergence results for approximations and domain-specific properties.

Findings

Birman-Schwinger operator is Hilbert-Schmidt on both domains.

The operator is not trace class.

Approximate operators converge to the original as the parameter tends to zero.

Abstract

We study the Birman-Schwinger operator for a self-adjoint realisation of the one-dimensional Hamiltonian with the Coulomb potential. We study both the case in which this Hamiltonian is defined on the whole real line and when it is only defined on the positive semiaxis. In both cases, the Birman-Schwinger operator is Hilbert-Schmidt, even though it is not trace class. Then, we have considered some approximations to the Hamiltonian depending on a positive parameter, under given conditions, and proved the convergence of the Birman-Schwinger operators of these approximations to the original Hamiltonian as the parameter goes to zero. Further comments and results have been included.