Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon

Abstract

In this paper we consider linear, time dependent Schrödinger equations of the form i∂tψ = K0ψ + V (t)ψ, where K0 is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and V (t) a time periodic potential. We give sufficient conditions on V (t) ensuring that K0 +V (t) generates unbounded orbits. The main condition is that the resonant average of V (t), namely the average with respect to the flow of K0, has a nonempty absolutely continuous spectrum and fulfills a Mourre estimate. These conditions are stable under perturbations. The proof combines pseudodifferential normal form with dispersive estimates in the form of local energy decay. We apply our abstract construction to the Harmonic oscillator on R and to the half-wave equation on T; in each case, we provide large classes of potentials which are transporters.

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

“…The proof of this result, that we will describe at the end of the section, build on the theory developed in [35], and it is based on a combination of pseudodifferential normal form and a dispersive mechanism in the energy space. The dispersion is quantitatively described by local energy decay estimates, which in turn are proved exploiting Mourre's theory of positive commutators.…”

“…To formalize this concept we shall say (following [35]) that V (t, x, D) is a transporter if (1.1) has at least one solution fulfilling (1.2) for some r > 0.…”

“…In the last few years several transporters for (1.1) were constructed by Delort [13], Bambusi-Grébert-M.-Robert [4], M. [34], Faou-Raphael [15], Liang, Zhao and Zhou [31], M. [35], Thomann [41], Luo, Liang and Zhao [32]; we will comment more about these results later on.…”

“…The dispersion is quantitatively described by local energy decay estimates, which in turn are proved exploiting Mourre's theory of positive commutators. We remark that in [35] we were already able to apply some abstract results to equation (1.1); however we were able only to deal with operators belonging to the special class of smooth Töplitz operators 1 . On the contrary, the main improvement of the current paper is to deal with generic classical pseudodifferential operators of order 0.…”

“…This shows that generic, classical pseudodifferential, 2π-periodic perturbations provoke unstable dynamics. The proof builds on the results of [35] and it is based on pseudodifferential normal form and local energy decay estimates. These last are proved exploiting Mourre's positive commutator theory.…”