Trajectory Symbols and the Fredholm Property of Boundary Value Problems for Differential Operators with Shifts

Abstract

Boundary value problems are considered in which the main operator and the operators of boundary conditions include differential and shift operators corresponding to the action of a discrete group. The manifold on which the boundary value problem is considered is not assumed to be group invariant. A definition of trajectory symbols for this class of boundary value problems is given. It is shown that elliptic problems define Fredholm operators in the corresponding Sobolev spaces. An application to problems with extensions and contractions is given.