Spectral Properties of the Dirac Operator coupled with $\delta$-Shell Interactions

Abstract

Let $\Omega \subset {\mathbb{R}}^3$ be an open set, we study the spectral properties of the free Dirac operator $\mathcal{H}$ coupled with the singular potential $V_{\kappa }=(\epsilon I_4+\mu \beta +\eta (\alpha \cdot N)){\delta }_{\partial \Omega }$. The open set $\Omega$ can be either a ${\mathcal{C}}^2$-bounded domain or a locally deformed half-space. In both cases, self-adjointness is proved and several spectral properties are given. In particular, we give a complete description of the essential spectrum of $\mathcal{H}+V_{\kappa }$ for the so-called critical combinations of coupling constants, when $\Omega$ is a locally deformed half-space. Finally, we introduce a new model of Dirac operators with $\delta$-interactions and deals with its spectral properties. More precisely, we study the coupling ${\mathcal{H}}_{\upsilon }=\mathcal{H}+i\upsilon \beta (\alpha \cdot N){\delta }_{\partial \Omega }$. In particular, we show that ${\mathcal{H}}_{\pm 2}$ is essentially self-adjoint and generates confinement.

This article corresponds to the first part of the article arXiv:2102.10207. In this version we corrected many typos and errors

References in corpus (4)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• General $\delta$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation
• Spectral Analysis of Dirac Operators with delta interactions supported on the boundaries of rough domains
• Spectral transition for Dirac operators with electrostatic $\delta$-shell potentials supported on the straight line

Cited by in corpus (5)

• Spectral Analysis of Dirac Operators with delta interactions supported on the boundaries of rough domains
• Eigenvalue curves for generalized MIT bag models
• Curvature contribution to the essential spectrum of Dirac operators with critical shell interactions
• A Poincaré-Steklov map for the MIT bag model
• On two-dimensional Dirac operators with critical delta-shell interactions