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Annales Henri Poincar´ e

Self-Adjoint Dirac Operators on Domains in R3 Jussi Behrndt , Markus Holzmann

and Albert Mas

Abstract. In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in L2 (Ω; C4 ), where Ω ⊂ R3 is either a bounded or an unbounded domain with a compact C 2 -smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with boundary conditions as of Robin type. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Kreintype resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman–Schwinger principle, a qualitative understanding of the scattering properties in the case that Ω is an exterior domain, and corresponding trace formulas. Mathematics Subject Classification. Primary 81Q10; Secondary 35Q40.

1. Introduction In recent years, the mathematical study of Dirac operators acting on domains Ω ⊂ Rd with special boundary conditions that make them self-adjoint gained a lot of attention. The motivation for this arises from several aspects: From the physical point of view, they are used in relativistic quantum mechanics to describe particles that are confined to a predefined area or box. One important model in 3D (dimension three) is the MIT bag model suggested in the 1970s by physicists in [30–32,34,43] to study confinement of quarks. In the 2D (dimension two) case, Dirac operators with special boundary conditions similar to the MIT bag model are used in the description of graphene; cf. [1,25,29,58]. From the mathematical point of view, Dirac operators with special boundary conditions can be seen as the relativistic counterpart of Laplacians with boundary conditions as, e.g., of Robin type. Moreover, Dirac operators with boundary

J. Behrndt et al.

Ann. Henri Poincar´e

conditions are also closely related to Dirac operators with singular δ–shell interactions supported on surfaces for special choices of the interaction strengths in the so-called confinement case, i.e., when the δ-potential is impenetrable for the particle; cf. [5,12,16,37]. To set the stage, let Ω ⊂ R3 be either a bounded or unbounded domain with a compact C 2 -smooth boundary and let ν be the unit normal vector field at ∂Ω which points outwards of Ω. Choose units such that the Planck constant  and the speed of light are both equal to one. Moreover, assume that ϑ : ∂Ω → R is a H¨older continuous function of order a > 12 , denoted by ϑ ∈ Lipa (∂Ω), and consider in L2 (Ω; C4 ) the operator Aϑ f = − iα · ∇f + mβf = −i

3 

αj · ∂j f + mβf,

j=1

      dom Aϑ = f ∈ H 1 (Ω; C4 ) : ϑ I4 + iβ(α · ν) f |∂Ω = I4 + iβ(α · ν) βf |∂Ω , (1.1) where α = (α1 , α2 , α3 ) and β are the C4×4 Dirac matrices defined in (1.6) and α · x = α1 x1 + α2 x2 + α3 x3 for x = (x1 , x2 , x3 ) ∈ R3 . The time-dependent equation with the Hamilt