Spectral Analysis of Dirac Operators with delta interactions supported on the boundaries of rough domains

TL;DR

This paper studies the spectral properties of Dirac operators with delta interactions on rough domain boundaries, focusing on self-adjointness, confinement, and regularity using layer potential techniques.

Contribution

It introduces a novel approach to analyze Dirac operators with boundary delta interactions on irregular domains, extending spectral theory in this context.

Findings

Established conditions for self-adjointness of the operators.

Analyzed confinement phenomena related to the boundary interactions.

Determined Sobolev regularity of the operator domains.

Abstract

Given an open set $\Omega\subset\mathbb{R}^3$. We deal with the spectral study of Dirac operators of the form $H_{a,\tau}=H+A_{a,\tau}\delta_{\partial\Omega}$, where $H$ is the free Dirac operator in $\mathbb{R}^3$, $A_{a,\tau}$ is a bounded invertible, self-adjoint operator in $\mathit{L}^{2}(\partial\Omega)^4$, depending on parameters $(a,\tau)\in\mathbb{R}\times\mathbb{R}^n$, $n\geqslant1$. We investigate the self-adjointness and the related spectral properties of $H_{a,\tau}$, such as the phenomenon of confinement and the Sobolev regularity of the domain in different situations. Our set of techniques, which is based on fundamental solutions and layer potentials, allows us to tackle the above problems under mild geometric measure theoretic assumptions on $\Omega$.