Dirac operators, shell interactions and discontinuous gauge functions across the boundary

TL;DR

This paper investigates the mathematical relationship between different types of shell interactions in the Dirac operator, showing their unitary equivalence and extending to magnetic shell potentials using explicit unitary transformations within gauge theory.

Contribution

It constructs explicit unitary operators to relate various shell interactions in the Dirac operator, including electrostatic and magnetic types, within a gauge-theoretic framework.

Findings

Unitary equivalence between perturbed and unperturbed electrostatic shell couplings.

Extension of unitary operators to magnetic shell potentials.

Explicit construction of operators for self-adjointness of shell interactions.

Abstract

Given a bounded smooth domain $\Omega\subset\mathbb{R}^3$, we explore the relation between couplings of the free Dirac operator $-i\alpha\cdot\nabla+m\beta$ with pure electrostatic shell potentials $\lambda\delta_{\partial\Omega}$ ($\lambda\in\mathbb{R}$) and some perturbations of those potentials given by the normal vector field $N$ on the shell $\partial\Omega$, namely $\{\lambda_e+\lambda_n(\alpha\cdot N)\}\delta_{\partial\Omega}$ ($\lambda_e$, $\lambda_n\in\mathbb{R}$). Under the appropiate change of parameters, the couplings with perturbed and unperturbed electrostatic shell potentials yield unitary equivalent self-adjoint operators. The proof relies on the construction of an explicit family of unitary operators that is well adapted to the study of shell interactions, and fits within the framework of gauge theory. A generalization of such unitary operators also allow us to deal with the self-adjointness of couplings of $-i\alpha\cdot\nabla+m\beta$ with some shell potentials of magnetic type, namely $\lambda(\alpha\cdot N)\delta_{\partial\Omega}$ with $\lambda\in C^1(\partial\Omega)$.