On Dirac operators in $\mathbb{R}^3$ with electrostatic and Lorentz scalar $δ$-shell interactions

Abstract

In this article Dirac operators $A_{η, τ}$ coupled with combinations of electrostatic and Lorentz scalar $δ$-shell interactions of constant strength $η$ and $τ$, respectively, supported on compact surfaces $Σ\subset \mathbb{R}^3$ are studied. In the rigorous definition of these operators the $δ$-potentials are modelled by coupling conditions at $Σ$. In the proof of the self-adjointness of $A_{η, τ}$ a Krein-type resolvent formula and a Birman-Schwinger principle are obtained. With their help a detailed study of the qualitative spectral properties of $A_{η, τ}$ is possible. In particular, the essential spectrum of $A_{η, τ}$ is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of $A_{η, τ}$ is computed and it is discussed that for some special interaction strengths $A_{η, τ}$ is decoupled to two operators acting in the domains with the common boundary $Σ$.