# On Dirac operators in $\mathbb{R}^3$ with electrostatic and Lorentz scalar $\delta$-shell interactions

## TL;DR

This paper rigorously analyzes Dirac operators with electrostatic and Lorentz scalar delta-shell interactions in three-dimensional space, establishing their spectral properties, self-adjointness, and nonrelativistic limits, with implications for quantum physics models.

## Contribution

It provides a rigorous mathematical framework for Dirac operators with delta-shell interactions, including self-adjointness, spectral analysis, and nonrelativistic limits, which was previously not fully developed.

## Key findings

- Essential spectrum explicitly determined
- Finitely many discrete eigenvalues can occur
- Symmetry relations in the point spectrum established

## Abstract

In this article Dirac operators $A_{\eta, \tau}$ coupled with combinations of electrostatic and Lorentz scalar $\delta$-shell interactions of constant strength $\eta$ and $\tau$, respectively, supported on compact surfaces $\Sigma \subset \mathbb{R}^3$ are studied. In the rigorous definition of these operators the $\delta$-potentials are modelled by coupling conditions at $\Sigma$. In the proof of the self-adjointness of $A_{\eta, \tau}$ a Krein-type resolvent formula and a Birman-Schwinger principle are obtained. With their help a detailed study of the qualitative spectral properties of $A_{\eta, \tau}$ is possible. In particular, the essential spectrum of $A_{\eta, \tau}$ is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of $A_{\eta, \tau}$ is computed and it is discussed that for some special interaction strengths $A_{\eta, \tau}$ is decoupled to two operators acting in the domains with the common boundary $\Sigma$.