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

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.

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.

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Accepté le :
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DOI : https://doi.org/10.5802/afst.1619
Mots clés : Reducibility, Quantum harmonic oscillator, KAM Theory
@article{AFST_2019_6_28_5_977_0,
     author = {Beno{\^\i}t Gr\'ebert and Eric Paturel},
     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 = {https://afst.centre-mersenne.org/articles/10.5802/afst.1619/}
}
Benoît Grébert; Eric Paturel. 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, Série 6, Tome 28 (2019) no. 5, pp. 977-1014. doi : 10.5802/afst.1619. https://afst.centre-mersenne.org/articles/10.5802/afst.1619/

[1] Pietro Baldi; Massimiliano Berti; Riccardo Montalto KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., Volume 359 (2014) no. 1-2, pp. 471-536 | Article | MR 3201904 | Zbl 1350.37076

[2] Dario Bambusi 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 | Article | Zbl 1145.37038

[3] Dario Bambusi Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations, II, Commun. Math. Phys., Volume 353 (2017) no. 1, pp. 353-378 | Article | Zbl 1367.35149

[4] Dario Bambusi Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations. I, Trans. Am. Math. Soc., Volume 370 (2018) no. 3, pp. 1823-1865 | Article | Zbl 1386.35058

[5] Dario Bambusi; Sandro Graffi Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Commun. Math. Phys., Volume 219 (2001) no. 2, pp. 465-480 | Article | Zbl 1003.37042

[6] Dario Bambusi; Benoît Grébert; Alberto Maspero; Didier Robert 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 1386.35059

[7] Jöran Bergh; Jörgen Löfström Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, Springer, 1976 | Zbl 0344.46071

[8] Nikolaĭ N. Bogoljubov; Yuriĭ A. Mitropoliskii; Anatoliĭ M. Samoĭlenko 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] Jean-Marc Delort; Jérémie Szeftel Long-time existence for small data nonlinear Klein–Gordon equations on tori and spheres, Int. Math. Res. Not., Volume 37 (2004), pp. 1897-1966 | Article | MR 2056326 | Zbl 1079.35070

[10] Håkan L. Eliasson 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 | Article | MR 1858550 | Zbl 1015.34028

[11] Håkan L. Eliasson; Sergei B. Kuksin On reducibility of Schrödinger equations with quasiperiodic in time potentials, Commun. Math. Phys., Volume 286 (2009) no. 1, pp. 125-135 | Article | Zbl 1176.35141

[12] Roberto Feola; Michela Procesi Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differ. Equations, Volume 259 (2015) no. 7, pp. 3389-3447 | Article | Zbl 1334.37087

[13] Benoît Grébert; Rafik Imekraz; Éric Paturel Normal forms for semilinear quantum harmonic oscillators, Commun. Math. Phys., Volume 291 (2009) no. 3, pp. 763-798 | Article | MR 2534791 | Zbl 1185.81073

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

[15] Benoît Grébert; Laurent Thomann KAM for the quantum harmonic oscillator, Commun. Math. Phys., Volume 307 (2011) no. 2, pp. 383-427 | Article | MR 2837120 | Zbl 1250.81033

[16] Bernard Helffer Théorie spectrale pour des opérateurs globalement elliptiques, Astérisque, Volume 112, Société Mathématique de France, 1984 (With an English summary) | Numdam | Zbl 0541.35002

[17] Àngel Jorba; Carles Simó On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differ. Equations, Volume 98 (1992) no. 1, pp. 111-124 | Article | MR 1168974 | Zbl 0761.34026

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

[19] Raphaël Krikorian Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts, Astérisque, Volume 259, Société Mathématique de France, 1999 | Numdam | MR 1732061 | Zbl 0957.37016

[20] Sergei B. Kuksin Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Mathematics, Volume 1556, Springer, 1993 | MR 1290785 | Zbl 0784.58028

[21] Jürgen Moser Convergent series expansions for quasi-periodic motions, Math. Ann., Volume 169 (1967), pp. 136-176 | Article | MR 208078 | Zbl 0149.29903

[22] Wei-Min Wang 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 | Article | MR 2358292 | Zbl 1144.81018