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Convergence of Schrödinger operators on domains with scaled resonant potentials

Vladimir Lotoreichik, Olaf Post

TL;DR

This work analyzes the convergence of Schrödinger operators on bounded smooth domains with potentials scaled in the normal direction to the boundary. A form-based framework, utilizing tubular coordinates and carefully constructed identification operators tied to resonant or non-resonant boundary-layer solutions, yields strong resolvent convergence results: resonant potentials produce a Robin Laplacian with a boundary coefficient given by the mean curvature, while non-resonant potentials (with a small negative part) converge to the Dirichlet Laplacian. The authors also establish that norm resolvent convergence cannot hold in general, providing a disk-based counterexample, and they discuss the role of mean curvature in the limiting boundary condition and the lack of uniform ellipticity in the non-resonant case. The results advance the understanding of boundary-interaction limits and provide a robust, kernel-free approach applicable to settings where the resolvent kernel is not explicitly known.

Abstract

We consider Schrödinger 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.

Convergence of Schrödinger operators on domains with scaled resonant potentials

TL;DR

This work analyzes the convergence of Schrödinger operators on bounded smooth domains with potentials scaled in the normal direction to the boundary. A form-based framework, utilizing tubular coordinates and carefully constructed identification operators tied to resonant or non-resonant boundary-layer solutions, yields strong resolvent convergence results: resonant potentials produce a Robin Laplacian with a boundary coefficient given by the mean curvature, while non-resonant potentials (with a small negative part) converge to the Dirichlet Laplacian. The authors also establish that norm resolvent convergence cannot hold in general, providing a disk-based counterexample, and they discuss the role of mean curvature in the limiting boundary condition and the lack of uniform ellipticity in the non-resonant case. The results advance the understanding of boundary-interaction limits and provide a robust, kernel-free approach applicable to settings where the resolvent kernel is not explicitly known.

Abstract

We consider Schrödinger operators on a bounded, smooth domain of dimension 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.
Paper Structure (17 sections, 15 theorems, 83 equations, 1 figure)

This paper contains 17 sections, 15 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Resonant potentials in one dimension
  4. Main results
  5. Preliminaries
  6. Tubular coordinates
  7. Quadratic forms and operators
  8. Convergence for resonant potentials
  9. Auxiliary boundary mappings
  10. The identification operator and an expression for the form difference
  11. Proof of Theorem \ref{['thm:main']}
  12. Convergence for non-resonant potentials
  13. Auxiliary boundary mappings, the identification operator and some related estimates
  14. Proof of Theorem \ref{['thm:main2']}
  15. No uniform ellipticity
  16. ...and 2 more sections

Key Result

Theorem 1

Assume that $V\in C^\infty_{\mathrm c}(\overline{{{\mathbb R}}}^{}_+)$ is resonant in the sense of Definition def:resonant. Then the family of scaled Schrödinger operators ${\mathsf A}_\varepsilon$ converges to the Robin Laplacian ${\mathsf A}$ in strong resolvent sense as $\varepsilon\to 0$.

Figures (1)

  • Figure 1.1: The solutions $\psi_0$ of \ref{['eq:IVP']} for resonant and non-resonant potentials on the left and on the right, respectively.

Theorems & Definitions (20)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1: the resonant case
  • Remark 1.3: appearance of mean curvature terms in related problems
  • Theorem 2: the non-resonant case
  • Remark 1.4: no uniform ellipticity
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 10 more