Index theorems for Fredholm, semi-Fredholm, and almost periodic operators: all in one example

Abstract

Based on operators borrowed from scattering theory, several concrete realizations of index theorems are proposed. The corresponding operators belong to some $C\ast$-algebras of pseudo-differential operators with coefficients which either have limits at $\pm \infty$, or which are periodic or asymptotically periodic, or which are uniformly almost periodic. These various situations can be deduced from a single partial isometry which depends on several parameters. All computations are explicitly performed.