Dirac operators, shell interactions, and discontinuous gauge functions across the boundary

Abstract

Abstract. Given a bounded smooth domain Ω ⊂ R 3 , we explore the relation between couplings of the free Dirac operator −iα · ∇ + mβ with pure electrostatic shell potentials λδ ∂Ω (λ ∈ R) and some perturbations of those potentials given by the normal vector field N on the shell ∂Ω, namely {λe + λn(α · N )}δ ∂Ω (λe, λn ∈ R). Under the appropiate change of parameters, the couplings with perturbed and unperturbed electrostatic shell potentials yield unitary equivalent self-adjoint operators. The proof relies on the construction of an explicit family of unitary operators that is well adapted to the study of shell interactions, and fits within the framework of gauge theory. A generalization of such unitary operators also allow us to deal with the self-adjointness of couplings of −iα · ∇ + mβ with some shell potentials of magnetic type, namely λ(α · N )δ ∂Ω with λ ∈ C 1 (∂Ω).

Self-Adjoint Dirac Operators on Domains in $$\mathbb {R}^3$$

“…Note that so far singularly perturbed Dirac operators of this form with electrostatic and Lorentz scalar δ-shell interactions have been studied only for constant coefficients η, τ ∈ R, see, e.g., [4][5][6]10,12,15,16,42,[49][50][51]55,56]. In particular, it is known that for constant η, τ ∈ R such that η 2 − τ 2 = −4 the operator B η,τ decouples into two self-adjoint operators acting in L 2 (Ω ± ; C 4 ) with certain boundary conditions; cf.…”

“…Assume that , let be real-valued functions on , and consider the formal differential expression where stands for the -distribution supported on the interface . In analogy to the case of constant interaction strengths in [ 12 , Section 3], we introduce the associated Dirac operator in as Note that so far singularly perturbed Dirac operators of this form with electrostatic and Lorentz scalar –shell interactions have been studied only for constant coefficients , see, e.g., [ 4 – 6 , 10 , 12 , 15 , 16 , 42 , 49 – 51 , 55 , 56 ]. In particular, it is known that for constant such that the operator decouples into two self-adjoint operators acting in with certain boundary conditions; cf.…”

“…However, this approach does not allow directly a systematic spectral analysis of these operators. Finally, we mention that in recent years the self-adjointness and the spectral and scattering properties of the closely related Dirac operators with singular δ-shell interactions were studied comprehensively in [4][5][6]10,12,15,16,42,[49][50][51]55,56].…”

Spectral Analysis of Dirac Operators with delta interactions supported on the boundaries of rough domains

“…where N is the (geometric measure theoretic) outward unit normal to Ω; see e.g. [2,29,5,30,31,34]. As it was observed in several works, for some particular values of the parameters the phenomenon of confinement arises, cf.…”

Eigenvalue curves for generalized MIT bag models

“…The family {H τ } τ ∈R naturally arises in the context of confining δ-shell interactions. In the last decade, Dirac operators coupled with singular δ-shell potentials have been investigated from a mathematical perspective: their self-adjointness and spectral properties [9,10,16,17,19,20,22,39,57,68], the case of rough domains [26,27], and their approximations and other asymptotic regimes [33,50,59,58,61]; we refer to the survey [63] for further details on the state of the art of shell interactions for Dirac operators. Several of these works addressed singular perturbations of the form…”