On the two-dimensional Coulomb-like potential with a central point interaction

TL;DR

This paper constructs and analyzes a two-dimensional Coulomb-like Hamiltonian with a central point interaction, exploring its properties and deriving it as a limit of a planar hydrogen atom model.

Contribution

It introduces a selfadjoint Hamiltonian with a point interaction for the 2D Coulomb potential and establishes its relation to the hydrogen atom in a thin slab.

Findings

Construction of a selfadjoint Hamiltonian with point interaction

Relation between coordinate and momentum representations

Derivation as a limit of planar hydrogen atom Hamiltonian

Abstract

In the first part of the paper, we introduce the Hamiltonian $-\Delta-Z/\sqrt{x^2+y^2}$, Z>0, as a selfadjoint operator in $L^2(R^2)$. A general central point interaction combined with the two-dimensional Coulomb-like potential is constructed and properties of the resulting one-parameter family of Hamiltonians is studied in detail. The construction is also reformulated in the momentum representation and a relation between the coordinate and the momentum representation is derived. In the second part of the paper we prove that the two-dimensional Coulomb-like Hamiltonian can be derived as a norm resolvent limit of the Hamiltonian of a Hydrogen atom in a planar slab as the width of the slab tends to zero.