On reducibility of quantum harmonic oscillator on d with quasiperiodic in time potential
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 5, pp. 977-1014.

We prove that a linear d-dimensional Schrödinger equation on d with harmonic potential |x| 2 and small t-quasiperiodic potential

itu-Δu+|x|2u+εV(tω,x)u=0,xd

reduces to an autonomous system for most values of the frequency vector ω n . As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in all Sobolev norms.

On montre que l’équation de Schrödinger d-dimensionnelle avec potentiel harmonique |x| 2 , perturbée par un petit potentiel quasipériodique en temps

itu-Δu+|x|2u+εV(tω,x)u=0,xd

est réductible à un système autonome pour la plupart des valeurs du vecteur de fréquences ω n . En conséquence, toute solution d’une telle EDP linéaire est presque-périodique en temps et toutes ses normes de Sobolev restent bornées.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1619
Keywords: Reducibility, Quantum harmonic oscillator, KAM Theory
Grébert, Benoît 1; Paturel, Eric 1

1 Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629, 2 rue de la Houssinière, 44322 Nantes Cedex 03, France
@article{AFST_2019_6_28_5_977_0,
     author = {Gr\'ebert, Beno{\^\i}t and Paturel, Eric},
     title = {On reducibility of quantum harmonic oscillator on $\protect \mathbb{R}^d$ with quasiperiodic in time potential},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {977--1014},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 28},
     number = {5},
     year = {2019},
     doi = {10.5802/afst.1619},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/afst.1619/}
}
TY  - JOUR
AU  - Grébert, Benoît
AU  - Paturel, Eric
TI  - On reducibility of quantum harmonic oscillator on $\protect \mathbb{R}^d$ with quasiperiodic in time potential
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2019
SP  - 977
EP  - 1014
VL  - 28
IS  - 5
PB  - Université Paul Sabatier, Toulouse
UR  - http://archive.numdam.org/articles/10.5802/afst.1619/
DO  - 10.5802/afst.1619
LA  - en
ID  - AFST_2019_6_28_5_977_0
ER  - 
%0 Journal Article
%A Grébert, Benoît
%A Paturel, Eric
%T On reducibility of quantum harmonic oscillator on $\protect \mathbb{R}^d$ with quasiperiodic in time potential
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2019
%P 977-1014
%V 28
%N 5
%I Université Paul Sabatier, Toulouse
%U http://archive.numdam.org/articles/10.5802/afst.1619/
%R 10.5802/afst.1619
%G en
%F AFST_2019_6_28_5_977_0
Grébert, Benoît; Paturel, Eric. On reducibility of quantum harmonic oscillator on $\protect \mathbb{R}^d$ with quasiperiodic in time potential. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 5, pp. 977-1014. doi : 10.5802/afst.1619. http://archive.numdam.org/articles/10.5802/afst.1619/

[1] Baldi, Pietro; Berti, Massimiliano; Montalto, Riccardo KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., Volume 359 (2014) no. 1-2, pp. 471-536 | DOI | MR | Zbl

[2] Bambusi, Dario A Birkhoff normal form theorem for some semilinear PDEs, Hamiltonian dynamical systems and applications (NATO Science for Peace and Security Series B: Physics and Biophysics), Springer, 2008, pp. 213-247 | DOI | Zbl

[3] Bambusi, Dario Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations, II, Commun. Math. Phys., Volume 353 (2017) no. 1, pp. 353-378 | DOI | Zbl

[4] Bambusi, Dario Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations. I, Trans. Am. Math. Soc., Volume 370 (2018) no. 3, pp. 1823-1865 | DOI | Zbl

[5] Bambusi, Dario; Graffi, Sandro Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Commun. Math. Phys., Volume 219 (2001) no. 2, pp. 465-480 | DOI | Zbl

[6] Bambusi, Dario; Grébert, Benoît; Maspero, Alberto; Robert, Didier Reducibility of the quantum harmonic oscillator in d-dimensions with polynomial time-dependent perturbation, Anal. PDE, Volume 11 (2018) no. 3, pp. 775-799 | Zbl

[7] Bergh, Jöran; Löfström, Jörgen Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, Springer, 1976 | Zbl

[8] Bogoljubov, Nikolaĭ N.; Mitropoliskii, Yuriĭ A.; Samoĭlenko, Anatoliĭ M. Methods of accelerated convergence in nonlinear mechanics, Hindustan Publishing Corp.; Springer, 1976 (Translated from the Russian by V. Kumar and edited by I. N. Sneddon)

[9] Delort, Jean-Marc; Szeftel, Jérémie Long-time existence for small data nonlinear Klein–Gordon equations on tori and spheres, Int. Math. Res. Not., Volume 37 (2004), pp. 1897-1966 | DOI | MR | Zbl

[10] Eliasson, Håkan L. Almost reducibility of linear quasi-periodic systems, Smooth ergodic theory and its applications (Seattle, WA, 1999) (Proceedings of Symposia in Pure Mathematics), Volume 69, American Mathematical Society, 2001, pp. 679-705 | DOI | MR | Zbl

[11] Eliasson, Håkan L.; Kuksin, Sergei B. On reducibility of Schrödinger equations with quasiperiodic in time potentials, Commun. Math. Phys., Volume 286 (2009) no. 1, pp. 125-135 | DOI | Zbl

[12] Feola, Roberto; Procesi, Michela Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differ. Equations, Volume 259 (2015) no. 7, pp. 3389-3447 | DOI | Zbl

[13] Grébert, Benoît; Imekraz, Rafik; Paturel, Éric Normal forms for semilinear quantum harmonic oscillators, Commun. Math. Phys., Volume 291 (2009) no. 3, pp. 763-798 | DOI | MR | Zbl

[14] Grébert, Benoît; Paturel, Éric KAM for the Klein Gordon equation on 𝕊 d , Boll. Unione Mat. Ital., Volume 9 (2016) no. 2, pp. 237-288

[15] Grébert, Benoît; Thomann, Laurent KAM for the quantum harmonic oscillator, Commun. Math. Phys., Volume 307 (2011) no. 2, pp. 383-427 | DOI | MR | Zbl

[16] Helffer, Bernard Théorie spectrale pour des opérateurs globalement elliptiques, Astérisque, 112, Société Mathématique de France, 1984 (With an English summary) | Numdam | Zbl

[17] Jorba, Àngel; Simó, Carles On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differ. Equations, Volume 98 (1992) no. 1, pp. 111-124 | DOI | MR | Zbl

[18] Koch, Herbert; Tataru, Daniel L p eigenfunction bounds for the Hermite operator, Duke Math. J., Volume 128 (2005) no. 2, pp. 369-392

[19] Krikorian, Raphaël Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts, Astérisque, 259, Société Mathématique de France, 1999 | Numdam | MR | Zbl

[20] Kuksin, Sergei B. Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Mathematics, 1556, Springer, 1993 | MR | Zbl

[21] Moser, Jürgen Convergent series expansions for quasi-periodic motions, Math. Ann., Volume 169 (1967), pp. 136-176 | DOI | MR | Zbl

[22] Wang, Wei-Min Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations, Commun. Math. Phys., Volume 277 (2008) no. 2, pp. 459-496 | DOI | MR | Zbl

Cited by Sources: