Convergence of Schr\"odinger operators on domains with scaled resonant potentials

TL;DR

This paper investigates how Schr"odinger operators with scaled, boundary-dependent potentials on smooth domains converge to different limit operators, revealing conditions for Robin or Dirichlet limits and highlighting the role of resonance.

Contribution

It provides a detailed analysis of the convergence behavior of Schr"odinger operators with scaled boundary potentials, distinguishing resonant and non-resonant cases and establishing new limit results.

Findings

Resonant scaled potentials lead to Robin Laplacian limits.

Non-resonant potentials with small negative parts lead to Dirichlet Laplacian limits.

Norm resolvent convergence does not hold in general for these operators.

Abstract

We consider Schr\"odinger operators on a bounded, smooth domain of dimension $d \ge 2$ with Dirichlet boundary conditions and a properly scaled potential, which depends only on the distance to the boundary of the domain. Our aim is to analyse the convergence of these operators as the scaling parameter tends to zero. If the scaled potential is resonant, the limit in strong resolvent sense is a Robin Laplacian with boundary coefficient expressed in terms of the mean curvature of the boundary. A counterexample shows that norm resolvent convergence cannot hold in general in this setting. If the scaled potential is non-resonant and satisfies an explicit assumption on the smallness of the negative part, the limit in strong resolvent sense is the Dirichlet Laplacian. We conjecture that we can drop this additional assumption in the non-resonant case.