Abstract
In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in L^2(Omega ; mathbb {C}^4), where Omega subset mathbb {R}^3 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 Krein-type 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 Omega is an exterior domain, and corresponding trace formulas.
Highlights
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 is the MIT bag model suggested in the 1970s by physicists in [30,31,32,34,43] to study confinement of quarks
Summary
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. Note that if {G, Γ0, Γ1} is a quasi boundary triple for T ⊂ S∗, Theorem 2.3 shows how the eigenvalues of self-adjoint extensions of S, which are contained in ρ(A0), can be characterized by the Weyl function M. Let S be a densely defined, closed, symmetric operator in H, let {G, Γ0, Γ1} be a quasi boundary triple for T ⊂ S∗, set A0 := T ker Γ0, and let γ and M be the associated γ-field and Weyl function, respectively. The unit normal vector field at ∂Ω pointing outwards Ω is denoted by ν
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