On Dirac operators with electrostatic δ-shell interactions of critical strength

Abstract

In this paper we prove that the Dirac operator $A_{\eta }$ with an electrostatic $\delta$-shell interaction of critical strength $\eta =\pm 2$ supported on a $C^2$-smooth compact surface $\Sigma$ is self-adjoint in $L^2({\mathbb{R}}^3;{\mathbb{C}}^4)$, we describe the domain explicitly in terms of traces and jump conditions in $H^{-1/2}(\Sigma ;{\mathbb{C}}^4)$, and we investigate the spectral properties of $A_{\eta }$. While the non-critical interaction strengths $\eta \ne \pm 2$ have received a lot of attention in the recent past, the critical case $\eta =\pm 2$ remained open. Our approach is based on abstract techniques in extension theory of symmetric operators, in particular, boundary triples and their Weyl functions.

28 pages, accepted for publication in J. Spectr. Theory

Cited by in corpus (1)

• The relativistic spherical $\delta$-shell interaction in ${\mathbb{R}}^3$: spectrum and approximation