Birkhoff resonances and quantum ray method

The known problem of resonances in the semiclassical approximation theory is analyzed. It is shown how a specific ??quantum behavior?? near the resonance motion appears. In the classical limit, we observe a Poisson micro-geometry concomitant with the resonance. We discuss what the corresponding quantum geometry is and construct irreducible representations of resonance algebras and their coherent states. This results in solving the resonance problems in question.