Growth of Sobolev norms in quasi integrable quantum systems

Abstract

We prove an abstract result giving a $⟨t{⟩}^{\epsilon }$ upper bound on the growth of the Sobolev norms of a time-dependent Schrödinger equation of the form $i\dot{\psi }=H_0\psi +V(t)\psi$. Here $H_0$ is assumed to be the Hamiltonian of a steep quantum integrable system and to be a pseudodifferential operator of order $\mathtt{d}>1$; $V(t)$ is a time-dependent family of pseudodifferential operators, unbounded, but of order $\mathtt{b}<\mathtt{d}$. 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. The proof is based on a quantum version of the proof of the classical Nekhoroshev theorem.

Cited by in corpus (2)

• Almost reducibility and oscillatory growth of Sobolev norms
• Growth of Sobolev norms for completely resonant quantum harmonic oscillators on ${\mathbb{R}}^2$