Fredholm property and essential spectrum of 3-D Dirac operators with regular and singular potentials

Abstract

ABSTRACT We consider the 3-D Dirac operator with variable regular magnetic and electrostatic potentials, and singular potentials (1) where (2) is the singular potential with being a matrix and is the delta-function with support on a surface which divides on two open domains with the common boundary Σ, is a vector-function on with values in are the standard Dirac matrices. We associate with the formal Dirac operator an unbounded operator in generated by with domain in consisting of functions satisfying transmission conditions on Σ. We consider the self-adjointness of operator , its Fredholm properties, and the essential spectrum in the case if Σ is either a closed -surface or an unbounded -hypersurface with a regular behaviour at infinity. As application we consider the electrostatic and Lorentz scalar δ-shell interactions.