Self-adjoint extensions of Coulomb systems in 1,2 and 3 dimensions

TL;DR

This paper investigates the self-adjoint extensions of the Coulomb Hamiltonian in one, two, and three dimensions, clarifying mathematical properties and addressing controversies about the origin's permeability.

Contribution

It characterizes all self-adjoint extensions of the Coulomb Hamiltonian in various dimensions and discusses the permeability of the origin in one dimension.

Findings

Complete classification of self-adjoint extensions in 1D, 2D, and 3D

Clarification of the controversy regarding the origin in 1D

Brief analysis of potentials from Laplace equation solutions

Abstract

We study the nonrelativistic quantum Coulomb hamiltonian (i.e., inverse of distance potential) in $R^n$, n = 1, 2, 3. We characterize their self-adjoint extensions and, in the unidimensional case, present a discussion of controversies in the literature, particularly the question of the permeability of the origin. Potentials given by fundamental solutions of Laplace equation are also briefly considered.