FREDHOLM PROPERTY OF INTERACTION PROBLEMS ON UNBOUNDED $$C^{2}-$$ HYPERSURFACES IN $$\mathbb{R}^{n}$$ FOR DIRAC OPERATORS

We consider the Dirac operators on $$\mathbb{R}^{n},n\ge 2$$ with singular potentials 1 $$\begin{aligned} D_{\varvec{A},\Phi ,m,\Gamma \delta _{\Sigma }}=\mathfrak {D}_{A,\Phi ,m}+\Gamma \delta _{\Sigma } \end{aligned}$$ where 2 $$\begin{aligned} \mathfrak {D}_{\varvec{A},\Phi ,m}=\sum \limits _{j=1}^{n}\alpha _{j}\left( -i\partial _{x_{j}}+A_{j}\right) +\alpha _{n+1}m+\Phi I_{N} \end{aligned}$$ is a Dirac operator on $$\mathbb{R}^{n}$$ with the variable magnetic and electrostatic potentials $$A\varvec{=}(A_{1},...,A_{n})$$ and $$\Phi$$ , and the variable mass m. In formula (2) $$\alpha _{j}$$ are the $$N\times N$$ Dirac matrices, that is $$\alpha _{j}\alpha _{k}+\alpha _{k}\alpha _{j}=2\delta _{jk}\mathbb{I}_{N}$$ , $$\mathbb{I}_{N}$$ is the unit $$N\times N$$ matrix, $$N=2^{\left[ \left( n+1\right) /2\right] },$$ $$\Gamma \delta _{\Sigma }$$ is a singular delta-type potential supported on a uniformly regular unbounded $$C^{2}-$$ hypersurface $$\Sigma \subset \mathbb{R}^{n}$$ being the common boundary of the open sets $$\Omega _{\pm }$$ . Let $$H^{1}(\Omega ^{\pm },\mathbb{C}^{N})$$ be the Sobolev spaces of $$\ N-$$ dimensional vector-valued distributions $$\varvec{u}$$ on $$\Omega ^{\pm },$$ and $$\begin{aligned} H^{1}(\mathbb{R}^{n}\diagdown \Sigma ,\mathbb{C}^{N})=H^{1}(\Omega _{+},\mathbb{C}^{N})\oplus H^{1}(\Omega _{-},\mathbb{C}^{N}). \end{aligned}$$ We associate with the formal Dirac operator $$D_{A,\Phi ,m,\Gamma \delta _{\Sigma }}$$ the interaction (transmission) operator $$\mathbb{D}_{A,\Phi ,m,B_{\Sigma }}=\left( \mathfrak {D}_{A,\Phi ,m},\mathfrak {B}_{\Sigma }\right)$$ defined by the Dirac operator $$\mathfrak {D}_{\varvec{A},\Phi ,m}$$ on $$H^{1}(\mathbb{R}^{n}\diagdown \Sigma ,\mathbb{C}^{N})$$ and the interaction condition $$\mathfrak {B}_{\Sigma }:$$ $$H^{1}(\mathbb{R}^{n}\mathbb{B}_{A,m},\Phi,\mathfrak{B}_{\Sigma}, \mathbb{C}^{N})\rightarrow H^{1/2}(\Sigma ,\mathbb{C}^{N})$$ associated with the singular potential. The main goal of the paper is to study the Fredholm property of the operators $$\mathbb{D}_{A,\Phi ,m,\mathfrak {B}_{\Sigma }}$$ for some non-compact $$C^{2}$$ -hypersurfaces.