Method of Potential Operators for Interaction Problems on Unbounded Hypersurfaces in $$\mathbb{R}^{n}$$ for Dirac Operators

Abstract

We consider the \(L_{p}\)-theory of interaction problems associated with Dirac operators with singular potentials of the form \(D=\mathfrak{D}_{m,\Phi }+\Gamma\delta_{\Sigma}\) where

$$\mathfrak{D}_{m,\Phi}=\sum_{j=1}^{n}\alpha_{j}(-i\partial_{x_{j}} )+m\alpha_{n+1}+\Phi\mathbb{I}_{N}$$

is a Dirac operator on \(\mathbb{R}^{n}\), \(\alpha_{1},\alpha_{2},\dots,\alpha _{n},\alpha_{n+1}\) are Dirac matrices, \(m\) is a variable mass, \(\Phi \mathbb{I}_{N}\) electrostatic potential, \(\Gamma\delta_{\Sigma}\) is a singular potential with support on smooth hypersurfaces \(\Sigma \subset\mathbb{R}^{n}.\)

We associate with the formal Dirac operator \(D\) the interaction (transmission) problem on \(\mathbb{R}^{n}\diagdown\Sigma\) with the interaction conditions on \(\Sigma\). Applying the method of potential operators we reduce the interaction problem to a pseudodifferential equation on \(\Sigma.\) The main aim of the paper is the study of Fredholm property of these pseudodifferential operators on unbounded hypersurfaces \(\Sigma\) and applications to the study of Fredholmness of interaction problems on unbounded smooth hypersurfaces in Sobolev and Besov spaces.

DOI 10.1134/S1061920823040167