Schrödinger Operators with Oblique Transmission Conditions in {mathbb R}^2

Abstract

In this paper we study the spectrum of self-adjoint Schrödinger 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 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 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.