Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type

Abstract

We present a detailed study in the semi classical regime h of microlocal properties of systems of two commuting h pseudo dierential operators Ph Ph such that the joint principal symbol p p p has a special kind of singularity called a focusfocus singularity Typical examples include the quantum spherical pendulum or the quantum Champagne bottle In the spirit of Colin de Verdiere and Parisse we show that such systems have a universal behavior described by singular quantization conditions of Bohr Sommerfeld type These conditions are used to give a precise description of the joint spectrum of such systems including the phenomenon of quantum monodromy and dierent formulations of the counting function for the joint eigenvalues close to the singularity in which a logarithm of the semi classical constant h appears Thanks to numerical computations done by MS Child for the case of the Champagne bottle we are able to accurately illustrate our statements