Growth of Sobolev norms in quasi integrable quantum systems

Abstract

We prove an abstract result giving a t ǫ upper bound on the growth of the Sobolev norms of a time dependent Schrödinger equation of the form i ψ = H 0 ψ + V (t)ψ. H 0 is assumed to be the Hamiltonian of a steep quantum integrable system and to be a pseudodifferential operator of order d > 1; V (t) is a time dependent family of pseudodifferential operators, unbounded, but of order b < d. The abstract theorem is then applied to perturbations of the quantum anharmonic oscillators in dimension 2 and to perturbations of the Laplacian on a manifold with integrable geodesic flow, and in particular Zoll manifolds, rotation invariant surfaces and Lie groups. We also cover the case of several particles on a Zoll manifold or on a Lie group, possibly obeying some restrictions due to the Fermionic or Bosonic nature of the particles. The proof is based a on quantum version of the proof of the classical Nekhoroshev theorem.

Generic transporters for the linear time dependent quantum Harmonic oscillator on $\mathbb R$

“…Energy cascade is a resonant phenomenon; here it happens because V (t, x, D) oscillates at frequency ω = 1 which resonates with the spectral gaps of the harmonic oscillator. In [4] we proved that if V (ωt, x, D) is quasiperiodic in time with a Diophantine frequency vector ω ∈ R n , then the Sobolev norms of the solutions grow at most as t ǫ , ∀ǫ > 0 (see [6] for recent results on t ǫ growth and reference therein). Moreover, with additional restriction on ω (typically belonging to some Cantor set of large measure) and assuming V (t, x, D) to be small in size, then all solutions have uniformly in time bounded Sobolev norms [3,5].…”

Almost Global Existence for Some Hamiltonian PDEs with Small Cauchy Data on General Tori

“…We emphasize that one of the points of interest of our paper is that it shows the impact of results of the kind of [15,20,25] dealing with linear time dependent systems on nonlinear systems, thus, in view of the generalizations [7][8][9], it opens the way to the possibility of proving almost global existence in more general systems, e.g. on some manifolds with integrable geodesic flow.…”