Pseudodifference operators on weighted spaces, and applications to discrete Schrodinger operators

We study pseudodifference operators on Z(N) with symbols which are bounded on Z(N) x T-N together with their derivatives with respect to the second variable. In the same way as partial differential operators on R-N are included in an algebra of pseudodifferential operators, difference operators on ZN are included in an algebra of pseudodifference operators. Particular attention is paid to the Fredholm properties of pseudodifference operators on general exponentially weighted spaces l(w)(p)(Z(N)) and to Phragmen - Lindelof type theorems on the exponential decay at infinity of solutions to pseudodifference equations. The results are applied to describe the essential spectrum of discrete Schrodinger operators and the decay of their eigenfunctions at infinity.