On the Energy Growth of Some Periodically Driven Quantum Systems with Shrinking Gaps in the Spectrum

Duclos, Pierre; Lev, Ondra; Šťovı́ček, P.

doi:10.1007/s10955-007-9419-5

Cited by 10 publications

(7 citation statements)

References 23 publications

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“…Upper bounds of the form (1.11) have a long history. The first results are due to Nenciu [Nen97], who proved, in an abstract framework, a xty ǫ upper bound for the expected value of the energy in case the system has increasing spectral gaps and bounded perturbation, and by Duclos, Lev and Šťovíček [DLS08] in case of decreasing spectral gaps.…”

Section: Growth Of Sobolev Norms For Linear Time Dependent Schrödinge...mentioning

confidence: 99%

Long time dynamics of Schrödinger and wave equations on flat tori

Berti

Maspero

2019

Journal of Differential Equations

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We consider a class of linear time dependent Schrödinger equations and quasi-periodically forced nonlinear Hamiltonian wave/Klein Gordon and Schrödinger equations on arbitrary flat tori. For the linear Schrödinger equation, we prove a t ǫ p@ǫ ą 0q upper bound for the growth of the Sobolev norms as the time goes to infinity. For the nonlinear Hamiltonian PDEs we construct families of time quasi-periodic solutions.Both results are based on "clusterization properties" of the eigenvalues of the Laplacian on a flat torus and on suitable "separation properties" of the singular sites of Schrödinger and wave operators, which are integers, in space-time Fourier lattice, close to a cone or a paraboloid. Thanks to these properties we are able to apply Delort abstract theorem [Del10] to control the speed of growth of the Sobolev norms, and Berti-Corsi-Procesi abstract Nash-Moser theorem [BCP15] to construct quasi-periodic solutions.

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Section: Growth Of Sobolev Norms For Linear Time Dependent Schrödinge...mentioning

confidence: 99%

Long time dynamics of Schrödinger and wave equations on flat tori

Berti

Maspero

2019

Journal of Differential Equations

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“…Duclos, Lev and Štovíček [5] have studied Sobolev bounds for solutions of such operators, under the same assumption of shrinking gap conditions. If one sets γ = 1−α 2 , and if the perturbation V (t) in (1) is periodic in time, small enough, and satisfies conditions of type…”

Section: Introductionmentioning

confidence: 99%

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

Delort

2013

Communications in Partial Differential Equations

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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.

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“…(which is violated if V (t) is 2π-periodic) then the Sobolev norms grow at most as t ǫ ∀ǫ > 0. The t ǫ -speed of growth is also known for systems with increasing [37,33,5] or shrinking [21,35] spectral gaps and for Schrödinger equation on T d with bounded [10,16,8] and even unbounded [7] potentials. 6.…”

Section: The Abstract Resultsmentioning

confidence: 99%

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

Maspero¹

2021

Preprint

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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.

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On the Energy Growth of Some Periodically Driven Quantum Systems with Shrinking Gaps in the Spectrum

Cited by 10 publications

References 23 publications

Long time dynamics of Schrödinger and wave equations on flat tori

Long time dynamics of Schrödinger and wave equations on flat tori

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

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

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