Semiclassical Normal Forms

Summary

Given a classical Hamiltonian function having an absolute minimum, we consider the problem of describing in the semiclassical limit the lowest part of the spectrum of the corresponding quantum operator. To this end we present an extension of the classical Birkhoff normal form to the semiclassical context and we use it to deduce spectral information on the quantum Hamiltonian. The properties of the spectrum turn out to be strongly dependent on the resonance relations fulfilled by the frequencies of small oscillations of the classical system. Here we concentrate on two opposite cases, namely the completely nonresonant and the completely resonant one and describe the spectrum of the Hamiltonian in these cases.