Spectral analysis of Dirac operators with delta interactions supported on the boundaries of rough domains

Abstract

Given an open set Ω⊂R3, we deal with the spectral study of Dirac operators of the form Ha,τ = H + Aa,τδ∂Ω, where H is the free Dirac operator in R3 and Aa,τ is a bounded, invertible, and self-adjoint operator in L2(∂Ω)4, depending on parameters (a,τ)∈R×Rn, n ⩾ 1. We investigate the self-adjointness and the related spectral properties of Ha,τ, such as the phenomenon of confinement and the Sobolev regularity of the domain in different situations. Our set of techniques, which is based on the use of fundamental solutions and layer potentials, allows us to tackle the above problems under mild geometric measure theoretic assumptions on Ω.