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We consider the Dirac operators on R n , n ≥ 2 with singular potentials 1 D A , Φ , m , Γ δ Σ = D A , Φ , m + Γ δ Σ where 2 D A , Φ , m = ∑ j = 1 n α j - i ∂ x j + A j + α n + 1 m + Φ I N is a Dirac operator on R n with the variable magnetic and electrostatic potentials A = (A 1 ,... , A n) and Φ , and the variable mass m. In formula (2) α j are the N × N Dirac matrices, that is α j α k + α k α j = 2 δ jk I N , I N is the unit N × N matrix, N = 2 n + 1 / 2 , Γ δ Σ is a singular delta-type potential supported on a uniformly regular unbounded C 2 - hypersurface Σ ⊂ R n being the common boundary of the open sets Ω ± . Let H 1 (Ω ± , C N) be the Sobolev spaces of N - dimensional vector-valued distributions u on Ω ± , and H 1 (R n ╲ Σ , C N) = H 1 (Ω + , C N) ⊕ H 1 (Ω - , C N). We associate with the formal Dirac operator D A , Φ , m , Γ δ Σ the interaction (transmission) operator D A , Φ , m , B Σ = D A , Φ , m , B Σ defined by the Dirac operator D A , Φ , m on H 1 (R n ╲ Σ , C N) and the interaction condition B Σ : H 1 (R n B A , m , Φ , B Σ , C N) → H 1 / 2 (Σ , C N) associated with the singular potential. The main goal of the paper is to study the Fredholm property of the operators D A , Φ , m , B Σ for some non-compact C 2 -hypersurfaces. [ABSTRACT FROM AUTHOR]

We consider the Dirac operators with singular potentials 1 D A , Φ , m , Γ δ Σ = D A , Φ , m + Γ δ Σ where 2 D A , Φ , m = ∑ j = 1 n α j - i ∂ x j + A j + α n + 1 m + Φ I N is a Dirac operator on R n with variable magnetic and electrostatic potentials A = (A 1 ,... , A n) , Φ , and the variable mass m. In formula (2), α j are the N × N Dirac matrices, that is α j α k + α k α j = 2 δ jk I N , I N is the unit N × N matrix, N = 2 n + 1 / 2 , Γ δ Σ is a singular delta-potential supported on C 2 - hypersurface Σ ⊂ R n periodic with respect to the action of a lattice G on R n. We consider the self-adjointnes and discretness of the spectrum of unbounded in L 2 (T , C N) operators associated with the formal Dirac operator (1) on the torus T = R N ∕ G . We study the band-gap structure of the spectrum of self-adjoint operators D in L 2 (R n , C N) associated with the formal Dirac operator (1) on R n with G -periodic regular and singular potentials. We also consider the Fredholm property and the essential spectrum of unbounded operators associated with non-periodic regular and singular potentials supported on G -periodic smooth hypersurfaces in R n. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R 3 be an open set. We study the spectral properties of the free Dirac operator H : = - i α · ∇ + m β coupled with the singular potential V κ = (ϵ I 4 + μ β + η (α · N)) δ ∂ Ω , where κ = (ϵ , μ , η) ∈ R 3 . The open set Ω can be either a C 2 -bounded domain or a locally deformed half-space. 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 + V κ 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 H ζ , υ = H + - i ζ α 1 α 2 α 3 + i υ β α · N δ ∂ Ω , with ζ , υ ∈ R . In particular, we show that H 0 , ± 2 is essentially self-adjoint and generates confinement. [ABSTRACT FROM AUTHOR]