Boundary triples and Weyl functions for Dirac operators with singular interactions

Abstract

In this article we develop a systematic approach to treat Dirac operators $A_{\eta, \tau, \lambda}$ with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions of strengths $\eta, \tau, \lambda \in \mathbb{R}$, respectively, supported on points in $\mathbb{R}$, curves in $\mathbb{R}^2$, and surfaces in $\mathbb{R}^3$ that is based on boundary triples and their associated Weyl functions. First, we discuss the one-dimensional case which also serves as a motivation for the multidimensional setting. Afterwards, in the two and three-dimensional situation we construct quasi, generalized, and ordinary boundary triples and their Weyl functions, and provide a detailed characterization of the associated Sobolev spaces, trace theorems, and the mapping properties of integral operators which play an important role in the analysis of $A_{\eta, \tau, \lambda}$. We make a substantial step towards more rough interaction supports $\Sigma$ and consider general compact Lipschitz hypersurfaces. We derive conditions for the interaction strengths such that the operators $A_{\eta, \tau, \lambda}$ are self-adjoint, obtain a Krein-type resolvent formula, and characterize the essential and discrete spectrum. These conditions include purely Lorentz scalar and purely non-critical anomalous magnetic interactions as well as the confinement case, the latter having an important application in the mathematical description of graphene. Using a certain ordinary boundary triple, we show the self-adjointness of $A_{\eta, \tau, \lambda}$ for arbitrary combinations of the interaction strengths (including critical ones) under the condition that $\Sigma$ is $C^{\infty}$-smooth and derive its spectral properties. In particular, in the critical case, a loss of Sobolev regularity in the operator domain and a possible additional point of the essential spectrum are observed.Comment: 56 pages; to appear in Reviews in Mathematical Physic