Analysis of high energy solutions of the Schrödinger equation for small perturbations of harmonic oscillators

In the context of Hamiltonian dynamical systems, Poincaré called the studying of near-integrable systems “the basic problem of the dynamics”. The most significant success in this area was the statement and proof by Kolmogorov, Arnold and Moser of the so called KAM theorem on persistence of quasiperiodic motions. Nowdays, KAM  theory studies several problems related to the stability and breakthrough of flow-invariant sets.

In this talk we will show how small perturbations of quantum harmonic oscillators modify the asymptotic behavior of solutions of the related Schrödinger equation, depending on the arithmetic relations between the frequencies of the oscillators.