  The Champagne bottle is one of the simplest and typical examples of Liouville
integrable systems that exhibit a non-trivial classical monodromy. This
geometrical invariant perturbs globally the existence of action-angle
coordinates on the phrase space. However, our work shows a new way to detect
the geometric modification on the phrase space by looking at the spectrum of a
single operator, that is small non-selfadjoint perturbation of a selfadjoint
semiclassical operator accepting the Hamiltonian of the Champagne bottle as its
principal symbol.
