Approximation of Schr\"odinger operators with point interactions on bounded domains

TL;DR

This paper develops a method to approximate point interactions in Schr"odinger operators on bounded domains using regular potentials, addressing a gap in the literature and exploring the role of zero energy resonances.

Contribution

It introduces an extension-restriction approach to approximate interior point interactions on bounded domains, filling a notable gap in existing research.

Findings

Approximation of point interactions via rescaled regular potentials.

Analysis of spectral assumptions and zero energy resonances.

Extension-restriction method applicable to bounded domains.

Abstract

We consider Schr\"odinger operators on a bounded domain $\Omega\subset \mathbb{R}^3$, with homogeneous Robin or Dirichlet boundary conditions on $\partial\Omega$ and a point (zero-range) interaction placed at an interior point of $\Omega$. We show that, under suitable spectral assumptions, and by means of an extension-restriction procedure which exploit the already known result on the entire space, the singular interaction is approximated by rescaled sequences of regular potentials. The result is missing in the literature, and we also take the opportunity to point out some general issues in the approximation of point interactions and the role of zero energy resonances.