On the spectral properties of Dirac operators with electrostatic $\delta$-shell interactions

TL;DR

This paper investigates the spectral, scattering, and asymptotic properties of Dirac operators with electrostatic delta-shell interactions, revealing finite discrete spectra, trace class resolvent differences, and nonrelativistic limits to Schrödinger operators.

Contribution

It introduces boundary triple techniques and resolvent formulas to analyze Dirac operators with delta-shell interactions, providing new insights into their spectral and scattering behavior.

Findings

Discrete spectrum inside the gap is finite

Difference of resolvents' third powers is trace class

Converges to Schrödinger operator in nonrelativistic limit

Abstract

In this paper the spectral properties of Dirac operators $A_\eta$ with electrostatic $\delta$-shell interactions of constant strength $\eta$ supported on compact smooth surfaces in $\mathbb{R}^3$ are studied. Making use of boundary triple techniques a Krein type resolvent formula and a Birman-Schwinger principle are obtained. With the help of these tools some spectral, scattering, and asymptotic properties of $A_\eta$ are investigated. In particular, it turns out that the discrete spectrum of $A_\eta$ inside the gap of the essential spectrum is finite, the difference of the third powers of the resolvents of $A_\eta$ and the free Dirac operator $A_0$ is trace class, and in the nonrelativistic limit $A_\eta$ converges in the norm resolvent sense to a Schr\"odinger operator with an electric $\delta$-potential of strength $\eta$.