Boundary triples and Weyl functions for Dirac operators with singular interactions

• Published in: Reviews in mathematical physics Vol. 36; no. 2
• Main Authors: Behrndt, Jussi, Holzmann, Markus, Stelzer-Landauer, Christian, Stenzel, Georg
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.03.2024; World Scientific Publishing Co. Pte., Ltd
• Subjects: Graphene; Hyperspaces; Magnetic properties; Operators (mathematics); Sobolev space; self-adjoint extension; Dirac operators with singular potentials; Weyl function; boundary triples
• ISSN: 0129-055X; 1793-6659
• DOI: 10.1142/S0129055X23500368

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.