Based on operators borrowed from scattering theory, we propose several concrete realisations of index theorems. The corresponding operators belong to some C*-algebras generated by bounded pseudo-differential operators with coefficients, which either have limits at ± ∞, or which are periodic, or which are asymptotically periodic. or which are uniform almost-periodic. These various situations can be deduced from a single partial isometry which depends on several parameters. The resulting relations corresponds to the topological version of Levison's theorem for a family of Schrödinger operators with inverse square potentials on the half-line. This talk is based on a joint work with S. Richard.