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

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