• Corpus ID: 250280153

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

@inproceedings{Behrndt2022SchrodingerOW,
  title={Schr\"odinger operators with oblique transmission conditions in \$\mathbb\{R\}^2\$},
  author={Jussi Behrndt and Markus Holzmann and Georg Stenzel},
  year={2022}
}

• J. Behrndt, M. Holzmann, Georg Stenzel
• Published 5 July 2022
• Mathematics

. In this paper we study the spectrum of self-adjoint Schr¨odinger operators in L 2 ( R 2 ) with a new type of transmission conditions along a smooth closed curve Σ ⊆ R 2 . Although these oblique transmission conditions are formally similar to δ ′ -conditions on Σ (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 δ -interactions justifying their usage as models in quantum mechanics.