Boundary triples and Weyl functions for Dirac operators with singular interactions

Abstract

In this paper, we develop a systematic approach to treat Dirac operators [Formula: see text] with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions of strengths [Formula: see text], respectively, supported on points in [Formula: see text], curves in [Formula: see text], and surfaces in [Formula: see text] 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 [Formula: see text]. We make a substantial step towards more rough interaction supports [Formula: see text] and consider general compact Lipschitz hypersurfaces. We derive conditions for the interaction strengths such that the operators [Formula: see text] 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 also show the self-adjointness of [Formula: see text] for arbitrary critical combinations of the interaction strengths under the condition that [Formula: see text] is [Formula: see text]-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.