Boundary triples and Weyl functions for Dirac operators with singular interactions

In this paper, we develop a systematic approach to treat Dirac operators A _η,τ,λ with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions of strengths η,τ,λ ∈ ℝ, respectively, supported on points in ℝ, curves in ℝ ², and surfaces in 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 _η,τ,λ. We make a substantial step towards more rough interaction supports Σ and consider general compact Lipschitz hypersurfaces. We derive conditions for the interaction strengths such that the operators A _η,τ,λ 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 A _η,τ,λ for arbitrary critical combinations of the interaction strengths under the condition that Σ is C ^∞-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.