Topological Levinson's theorem in presence of embedded thresholds and discontinuities of the scattering matrix

Abstract: A family of discrete Schroedinger operators is investigated through scattering theory. The continuous spectrum of these operators exhibit changes of multiplicity, and some of these operators possess resonances at thresholds. It is shown that the corresponding wave operators belong to an explicitly constructed C*-algebra, whose K-theory is carefully analysed. An index theorem is deduced from these investigations, which corresponds to a topological version of Levinson's theorem in presence of embedded thresholds, resonances, and changes of multiplicity of the scattering matrices. In the second half of the paper, very detailed computations for the simplest realisation of this family of operators are provided. In particular, a surface of resonances is exhibited, probably for the first time. For Levinson's theorem, it is shown that contributions due to resonances at the lowest value and at the highest value of the continuous spectrum play an essential role.