Schrödinger Operators with Oblique Transmission Conditions in R2

Abstract

In this paper we study the spectrum of self-adjoint Schrödinger operators in $L^2\left(R^2\right)$ with a new type of transmission conditions along a smooth closed curve $\Sigma \subseteq R^2$. Although these oblique transmission conditions are formally similar to ${\delta }^{\text{'}}$-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 Schrödinger 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.