Dirac Operators with Delta-Interactions on Smooth Hypersurfaces in Rⁿ

We consider the Dirac operators with singular potentials DA,Φ,m,ΓδΣ=DA,Φ,m+ΓδΣwhere DA,Φ,m=∑j=1nαj(-i∂xj+Aj)+αn+1m+ΦINis a Dirac operator on Rⁿ with variable magnetic and electrostatic potentials A=(A₁, … , A_n) ∈ L^∞(Rⁿ, Cⁿ) and Φ ∈ L^∞(Rⁿ) , and the variable mass m∈ L^∞(Rⁿ). In formula (2) α_j are the N× N Dirac matrices, that is α_jα_k+ α_kα_j= 2 δ_jkI_N, I_N is the unit N× N matrix, N= 2 ^{[ ( n + 1 ) / 2 ]}. In formula (1) Γ δ_Σ is a singular delta-type potential supported by a C²-hypersurface Σ ⊂ Rⁿ which is the common boundary of the open sets Ω _±. Let H¹(Ω ^±, C^N) be the Sobolev spaces of N-dimensional vector-valued distributions u on Ω ^±, and H1(Rn╲Σ,CN)=H1(Ω+,CN)⊕H1(Ω-,CN).We associate with the formal Dirac operator DA,Φ,m,ΓδΣ an unbounded in L²(Rⁿ, C^N) operator DA,Φ,m,BΣ defined by the Dirac operator D_{A , Φ , m} with domain domDA,Φ,m,BΣ⊂H1(Rn╲Σ,CN) defined by an interaction conditions. The main aims of the paper are the study of self-adjointmess of the operators DA,Φ,m,BΣ for uniformly regular C²-hypersurfaces Σ ⊂ Rⁿ and the essential spectra of DA,Φ,m,BΣ for closed C²-hypersurfaces Σ ⊂ Rⁿ.