Spectral Properties of the Dirac Operator coupled with $\delta$-Shell Interactions

Let Ω ⊂ R be an open set. We study the spectral properties of the free Dirac operator H := −iα · ∇ + mβ coupled with the singular potential Vκ = (ǫI4 + μβ + η(α · N))δ∂Ω, where κ = (ǫ, μ, η) ∈ R. The open set Ω can be either a C-bounded domain or a locally deformed halfspace. In both cases, self-adjointness is proved and several spectral properties are given. In particular, we give a complete description of the essential spectrum of H + Vκ in the case of a locally deformed half-space, for the so-called critical combinations of coupling constants. Finally, we introduce a new model of Dirac operators with δ-interactions and deal with its spectral properties. More precisely, we study the coupling Hζ,υ = H + (−iζα1α2α3 + iυβ (α ·N)) δ∂Ω, with ζ, υ ∈ R. In particular, we show that H0,±2 is essentially self-adjoint and generates confinement.