Mathematical predominance of Dirichlet condition for the one-dimensional Coulomb potential

TL;DR

This paper investigates the limiting behavior of a quantum particle under Coulomb potential confined in shrinking tubes, demonstrating that the Dirichlet boundary condition naturally emerges as the limit in both attractive and repulsive cases.

Contribution

It proves that the Dirichlet boundary condition at the origin arises as the natural limit for the Schrödinger operator with Coulomb potential in a confining tube setting.

Findings

Strong resolvent convergence to Dirichlet operator in repulsive case

Norm resolvent convergence in regularized attractive case

Dirichlet boundary condition is the natural limit among self-adjoint realizations

Abstract

We restrict a quantum particle under a coulombian potential (i.e., the Schr\"odinger operator with inverse of the distance potential) to three dimensional tubes along the x-axis and diameter $\varepsilon$, and study the confining limit $\varepsilon\to0$. In the repulsive case we prove a strong resolvent convergence to a one-dimensional limit operator, which presents Dirichlet boundary condition at the origin. Due to the possibility of the falling of the particle in the center of force, in the attractive case we need to regularize the potential and also prove a norm resolvent convergence to the Dirichlet operator at the origin. Thus, it is argued that, among the infinitely many self-adjoint realizations of the corresponding problem in one dimension, the Dirichlet boundary condition at the origin is the reasonable one-dimensional limit.