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Two-dimensional Schrödinger operators with non-local singular potentials

Lukáš Heriban, Markus Holzmann, Christian Stelzer-Landauer, Georg Stenzel, Matěj Tušek

TL;DR

This work develops a rigorous extension-theoretic framework for two-dimensional Schrödinger operators with non-local singular potentials supported on a Lipschitz curve $Σ$, yielding a self-adjoint operator $T_B$ parameterized by compact Hermitian $B$ on $L^2(Σ;\mathbb{C}^2)$. Using a tailored generalized boundary triple, the authors derive a Krein-type resolvent formula and a Birman-Schwinger principle to characterize the spectrum, showing a stable essential spectrum $[0,\infty)$ and a richly structured discrete spectrum ranging from finite to infinite with accumulation at $-\infty$ under suitable $B$. They compare these non-local transmission conditions to classical $\,δ$-shell and oblique transmissions, and establish an explicit non-relativistic limit linking the Schrödinger model to a relativistic, non-local Dirac model studied in HT23. Overall, the paper provides a versatile operator-analytic framework for non-local surface interactions with concrete spectral consequences and a clear relativistic correspondence.

Abstract

In this paper we introduce and study a family of self-adjoint realizations of the Laplacian in $L^2(\mathbb{R}^2)$ with a new type of transmission conditions along a closed bi-Lipschitz curve $Σ$. These conditions incorporate jumps in the Dirichlet traces both of the functions in the operator domains and of their Wirtinger derivatives and are non-local. Constructing a convenient generalized boundary triple, they may be parametrized by all compact hermitian operators in $L^2(Σ;\mathbb{C}^2)$. Whereas for all choices of parameters the essential spectrum is stable and equal to $[0, +\infty)$, the discrete spectrum exhibits diverse behaviour. While in many cases it is finite, we will describe also a class of parameters for which the discrete spectrum is infinite and accumulates at $-\infty$. The latter class contains a non-local version of the oblique transmission conditions. Finally, we will connect the current model to its relativistic counterpart studied recently in [L. Heriban, M. Tušek: Non-local relativistic $δ$-shell interactions].

Two-dimensional Schrödinger operators with non-local singular potentials

TL;DR

This work develops a rigorous extension-theoretic framework for two-dimensional Schrödinger operators with non-local singular potentials supported on a Lipschitz curve , yielding a self-adjoint operator parameterized by compact Hermitian on . Using a tailored generalized boundary triple, the authors derive a Krein-type resolvent formula and a Birman-Schwinger principle to characterize the spectrum, showing a stable essential spectrum and a richly structured discrete spectrum ranging from finite to infinite with accumulation at under suitable . They compare these non-local transmission conditions to classical -shell and oblique transmissions, and establish an explicit non-relativistic limit linking the Schrödinger model to a relativistic, non-local Dirac model studied in HT23. Overall, the paper provides a versatile operator-analytic framework for non-local surface interactions with concrete spectral consequences and a clear relativistic correspondence.

Abstract

In this paper we introduce and study a family of self-adjoint realizations of the Laplacian in with a new type of transmission conditions along a closed bi-Lipschitz curve . These conditions incorporate jumps in the Dirichlet traces both of the functions in the operator domains and of their Wirtinger derivatives and are non-local. Constructing a convenient generalized boundary triple, they may be parametrized by all compact hermitian operators in . Whereas for all choices of parameters the essential spectrum is stable and equal to , the discrete spectrum exhibits diverse behaviour. While in many cases it is finite, we will describe also a class of parameters for which the discrete spectrum is infinite and accumulates at . The latter class contains a non-local version of the oblique transmission conditions. Finally, we will connect the current model to its relativistic counterpart studied recently in [L. Heriban, M. Tušek: Non-local relativistic -shell interactions].

Paper Structure

This paper contains 12 sections, 27 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Curves, function spaces, and trace theorems
  5. Distributions
  6. Integral operators
  7. Generalized boundary triples
  8. Definition and spectral properties of $T_B$
  9. A generalized boundary triple suitable for the study of $T_B$
  10. Definition and Self-adjointness of $T_B$
  11. Spectrum of $T_B$
  12. The Dirac operator with a non-local singular potential and its non-relativistic limit

Key Result

Lemma 2.2

Let $s \in [\frac{1}{2},\frac{3}{2}]$. Then the mapping $C_0^{\infty}(\overline{\Omega_{\pm}}) \ni f \mapsto f |_{\Sigma}$ can be extended to linear and bounded operators Furthermore, for all $f \in H_{\partial_{\bar{z}}}^{s}(\Omega_{\pm})$ and all $g \in H_{\partial_z}^{s}(\Omega_{\pm})$ there holds If $\Sigma$ is a $C^{k}$-boundary with $k \geq 2$ then the above mapping properties remain vali

Theorems & Definitions (53)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 43 more