Classical and Quantum Perturbation Theory for two Non--Resonant   Oscillators with Quartic Interaction
  We study the classical and quantum perturbation theory for two non--resonant oscillators coupled by a nonlinear quartic interaction. In particular we analyze the question of quantum corrections to the torus quantization of the classical perturbation theory (semiclassical mechanics). We obtain up to the second order of perturbation theory an explicit analytical formula for the quantum energy levels, which is the semiclassical one plus quantum corrections. We compare the "exact" quantum levels obtained numerically to the semiclassical levels studying also the effects of quantum corrections. 
Luca Salasnich