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 $\text{oblique}$ transmission conditions are formally similar to ${\delta }^{\prime }$-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.

16 pages; to appear in Comm. Math. Phys

References in corpus (1)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces

Cited by in corpus (1)

• The $\overline{\partial }$-Robin Laplacian