KAM for the Quantum Harmonic Oscillator

Grébert, Benoît; Thomann, Laurent

doi:10.1007/s00220-011-1327-5

Cited by 98 publications

(126 citation statements)

References 16 publications

Supporting

Mentioning

118

Contrasting

Unclassified

Order By: Relevance

“…We do not know if a result of type (1.1.6) could be proved replacing in (1.1.5) the order zero perturbation Op W (V (t, ·)) by a local potential V (t, x). On the other hand, Grébert and Thomann [8] have studied recently equations of the form (1.1.5), where V is a small time quasi-periodic local potential satisfying convenient assumptions. They have been able to prove that, when the parameter of quasi-periodicity stays outside a subset of small measure, the equation may be reduced to a linear equation with constant coefficients.…”

Section: Remarksmentioning

confidence: 99%

“…Consequently, the solutions of the Cauchy problem are time quasi-periodic, and so have uniformly bounded Sobolev norms. We refer to theorem 1.2 and corollary 1.3 of [8] for precise statements, and to corollary 1.4 of the same paper, as well as to the paper of Wang [13], for interpretation of such results in terms of Floquet spectrum. Notice also that similar results for the Schrödinger equation on the torus, with time quasi-periodic potential, had been previously proved by Eliasson and Kuksin [6].…”

Section: Remarksmentioning

confidence: 99%

See 1 more Smart Citation

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

Delort

2013

Communications in Partial Differential Equations

View full text Add to dashboard Cite

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.

show abstract

Section: Remarksmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

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

Delort

2013

Communications in Partial Differential Equations

View full text Add to dashboard Cite

show abstract

“…Similar as [22], the above theorem has two direct corollaries. As a preparation we define the harmonic oscillator T = − d 2 dx 2 + x 2 and its related Sobolev space.…”

Section: Statement Of the Resultsmentioning

confidence: 78%

Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay

Wang

Liang

2017

Nonlinearity

View full text Add to dashboard Cite

Abstract. In this paper we prove an infinite dimensional KAM theorem, in which the assumptions on the derivatives of perturbation in [22] are weakened from polynomial decay to logarithmic decay. As a consequence, we apply it to 1d quantum harmonic oscillators and prove the reducibility of a linear harmonic oscillator,perturbed by a quasiperiodic in time potential V (x, ωt; ω) with logarithmic decay. This entails the pure-point nature of the spectrum of the Floquet operator K, whereand the potential V (x, θ; ω) has logarithmic decay as well as its gradient in ω.

show abstract

“…All these examples concern PDEs on the torus, essentially because in that case the corresponding linear PDE is diagonalized in the Fourier basis and the structure of the resonant sets remains almost the same. Recently I have considered (see [19]) two important examples that do not fit in this Fourier context: the Klein-Gordon equation on the sphere S 2 and the quantum harmonic oscillator on R 2 . In both cases I use external parameters.…”

Section: Short Review Of Related Literaturementioning

confidence: 99%

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

Section: Short Review Of Related Literaturementioning

confidence: 99%

Recent results on KAM for multidimensional PDEs

Grébert¹

2014

Journées Équations Aux Dérivées Partielles

Self Cite

View full text Add to dashboard Cite

In this short overview I present some recent results about the KAM theory for multidimensional partial differential equations (PDEs) trying to avoid technicalities. In particular I will not state a precise KAM theorem but I will focus on the dynamical consequences for the PDEs: the existence and the stability (or not) of quasi periodic in time solutions. Concretely, I present the complete study of the nonlinear beam equation on the d-dimensional torus recently obtained in collaboration with H. Eliasson and S. Kuksin. When d ≥ 2 we are able to construct explicit examples where the quasi periodic solutions are linearly unstable, a new feature in Hamiltonian PDEs that could complement recent results in weak turbulence theory.

show abstract

KAM for the Quantum Harmonic Oscillator

Cited by 98 publications

References 16 publications

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

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

Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay

Recent results on KAM for multidimensional PDEs

Contact Info

Product

Resources

About