Dirac operators with Lorentz scalar shell interactions

TL;DR

This paper analyzes the spectral properties of a massive Dirac operator with Lorentz scalar shell interactions on a smooth surface, revealing eigenvalue behavior influenced by boundary Schrödinger operators with gauge and curvature effects.

Contribution

It provides a rigorous definition, proves self-adjointness, and investigates eigenvalue existence and asymptotics for the Dirac operator with scalar shell interactions.

Findings

Eigenvalues governed by boundary Schrödinger operator with Yang-Mills potential

Asymptotic behavior of eigenvalues for large mass analyzed

Conditions for eigenvalue existence and non-existence established

Abstract

This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. After showing the self-adjointness of the resulting operator we switch to the investigation of its spectral properties, in particular, to the existence and non-existence of eigenvalues. In the case of an attractive coupling, we study the eigenvalue asymptotics as the mass becomes large and show that the behavior of the individual eigenvalues and their total number are governed by an effective Schr\"odinger operator on the boundary with an external Yang-Mills potential and a curvature-induced potential.