Growth of Sobolev Norms for Solutions of Time Dependent Schrödinger Operators with Harmonic Oscillator Potential

Abstract

It has been known for some time that solutions of linear Schrödinger operators on the torus, with bounded, smooth, time dependent (order zero pseudo-differential) potential, have Sobolev norms growing at most like t when t → +∞ for any > 0. This property is proved exploiting the fact that, on the circle, successive eigenvalues of the laplacian are separated by increasing gaps (and a more involved, but similar property, for clusters of eigenvalues in higher dimension). We study here the case of solutions ofwhere V is a time periodic pseudo-differential order zero perturbation. In this case, the gap between successive eigenvalues of the stationary operator is constant. We show that there are order zero potentials V for which some solutions u have Sobolev norms of order s growing like t s/2 when t → +∞, i.e. lim infThe idea of the proof is to construct a potential which, at the classical level, pulls frequencies to higher modes, so that they will be of size √ t at time t. One contructs then the wanted solution passing from the classical level to the quantum one. IntroductionLet P 0 be an elliptic self-adjoint differential operator of positive order on some Riemannian manifold M . Consider t → V (t) a smooth family of self-adjoint operators on M , or order zero, and the Schrödinger equationDenote by H s the Sobolev space associated to P 0 , defined when s is an even integer 2kWe are interested in estimating u(t, ·) H s when t goes to infinity. If V is time independent, it is immediate that u(t, ·) H s = u(0, ·) H s . Over the last fifteen years, several results have been obtained by different authors for time dependent potentials V , under convenient spectral assumptions on P 0 and V . Our aim is to This work was partially supported by the ANR project Equa-disp.

Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations

“…This is a slightly weaker statement than (2); however, in applications, one can typically prove the stronger estimate e −itA ψ r ≥ C r,ψ t r as t → ∞. In particular we succeed in doing this for the Harmonic oscillator on R, giving an alternative, shorter proof of Delort's result [Del14] (see Theorem 3.4).…”

“…on the contrary, in case (iii) V (t) has to contain a not perturbative term to correct the spectral gaps. While the techniques of [Bou99,Del14] are quite involved, the construction of [BGMR18] is simpler and based on a result by Graffi and Yajima [GY00] to prove stability of the absolutely continuous spectrum of a certain Floquet operator.…”

Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

“…The reason is that for system (1.1) with l = 1 and V as in (1.11) the classical-semiclassical correspondence is exact and valid for all times, a property first exploited by Enss and Veselić [15]. This is also the mechanism exploited in [3], which ultimately is based on the fact that (1.9) with l = 1 and β(t) = sin t is a resonant system, whose solutions are unbounded (see also [13,30] for different examples of perturbations provoking growth of Sobolev norms). In case l ≥ 2, the classical-semiclassical correspondence is valid only for finite times, and the speed of growth of Sobolev norms is logarithmic and not polynomial in time.…”

Recent results on KAM for multidimensional PDEs

“…And a related question: Is the flow bounded in Sobolev spaces? (see [13,20,9] for a first overview on this problem).…”