Topological Levinson's theorem and corrections at thresholds: the full picture in a quasi-1D example

Abstract: Various threshold effects are investigated on a discrete quasi-1D scattering system. In particular, one of these effects is to add corrections to Levinson's theorem. We explain how these corrections are due to the opening or to the closing of channels of scattering, and how these contributions can be computed as partial winding numbers on newly introduced operators. Embedded thresholds, thresholds associated with changes of spectral multiplicity, and doubly degenerate thresholds are exhibited and analyzed. Most of the investigations are of an analytical nature, but the final equalities rely on a C*-algebraic framework.