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

Based on operators borrowed from scattering theory, several concrete realizations of index theorems are proposed. The corresponding operators belong to some C*-algebras of pseudo-differential operators with coefficients which either have limits at +infinity, 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.