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 Schrödinger operator with an electric $\delta$-potential of strength $\eta$.

32 pages

References in corpus (4)

• Dirac materials
• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Approximation of Schrödinger operators with $\delta$-interactions supported on hypersurfaces

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• Schrödinger operators with oblique transmission conditions in ${\mathbb{R}}^2$
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