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

• Published in: Journal of mathematical sciences (New York, N.Y.) N.Y.), 2023, Vol.271 (2), p.136-161
• Main Author: Rabinovich, Vladimir S.
• Format: Article
• Language: English
• Subjects: Hyperspaces; Mathematics; Mathematics and Statistics; Operators (mathematics); Sobolev space
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-023-06326-z

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.