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

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 $\eta$.