Two-dimensional Schr\"odinger operators with non-local singular potentials

TL;DR

This paper introduces a new family of self-adjoint Laplacian operators in two dimensions with non-local transmission conditions along a curve, analyzing their spectral properties and connecting to relativistic models.

Contribution

It develops a novel framework for Laplacians with non-local boundary conditions on curves, including a parametrization and spectral analysis, extending previous local models.

Findings

Essential spectrum remains stable at [0, +∞)

Discrete spectrum can be finite or infinite depending on parameters

Infinite discrete spectrum can accumulate at -∞

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 $\Sigma$. 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 $L^2(\Sigma;\mathbb{C}^2)$. 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. Tu\v{s}ek: Non-local relativistic $\delta$-shell interactions].