Schr\"odinger operators with oblique transmission conditions in $\mathbb{R}^2$

TL;DR

This paper investigates a new class of Schr"odinger operators with oblique transmission conditions, revealing unique spectral properties and establishing their connection to Dirac operators in quantum mechanics.

Contribution

It introduces and analyzes Schr"odinger operators with oblique transmission conditions, showing their spectral behavior and deriving key formulas, linking them to Dirac operators as non-relativistic limits.

Findings

Discrete spectrum unbounded from below for attractive interactions

Identified the essential spectrum of the operators

Derived Krein-type resolvent formula and Birman-Schwinger principle

Abstract

In this paper we study the spectrum of self-adjoint Schr\"odinger operators in $L^2(\mathbb{R}^2)$ with a new type of transmission conditions along a smooth closed curve $\Sigma\subseteq \mathbb{R}^2$. Although these $\textit{oblique}$ transmission conditions are formally similar to $\delta'$-conditions on $\Sigma$ (instead of the normal derivative here the Wirtinger derivative is used) the spectral properties are significantly different: it turns out that for attractive interaction strengths the discrete spectrum is always unbounded from below. Besides this unexpected spectral effect we also identify the essential spectrum, and we prove a Krein-type resolvent formula and a Birman-Schwinger principle. Furthermore, we show that these Schr\"odinger operators with oblique transmission conditions arise naturally as non-relativistic limits of Dirac operators with electrostatic and Lorentz scalar $\delta$-interactions justifying their usage as models in quantum mechanics.