FREDHOLM PROPERTY OF INTERACTION PROBLEMS ON UNBOUNDED C2- HYPERSURFACES IN Rn FOR DIRAC OPERATORS

Abstract

We consider the Dirac operators on Rn, n≥ 2 with singular potentials DA,Φ,m,ΓδΣ=DA,Φ,m+ΓδΣ where DA,Φ,m=∑j=1nαj(-i∂xj+Aj)+αn+1m+ΦIN is a Dirac operator on Rn with the variable magnetic and electrostatic potentials A=(A1,.. , An) and Φ , and the variable mass m. In formula (2) αj are the N× N Dirac matrices, that is αjαk+ αkαj= 2 δjkIN , IN is the unit N× N matrix, N= 2 [(n+1)/2], Γ δΣ is a singular delta-type potential supported on a uniformly regular unbounded C2- hypersurface Σ ⊂ Rn being the common boundary of the open sets Ω ± . Let H1(Ω ±, CN) 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 the interaction (transmission) operator DA,Φ,m,BΣ=(DA,Φ,m,BΣ) defined by the Dirac operator DA,Φ,m on H1(Rn╲ Σ , CN) and the interaction condition BΣ: H1(RnBA,m, Φ , BΣ, CN) → H1 / 2(Σ , CN) associated with the singular potential. The main goal of the paper is to study the Fredholm property of the operators DA,Φ,m,BΣ for some non-compact C2 -hypersurfaces.