  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.
