Growth of Sobolev norms for linear Schrödinger operators
Annales Henri Lebesgue, Volume 4 (2021), pp. 1595-1618.

We give an example of a linear, time-dependent, Schrödinger operator with optimal growth of Sobolev norms. The construction is explicit, and relies on a comprehensive study of the linear Lowest Landau Level equation with a time-dependent potential.

Nous donnons un exemple d’un opérateur de Schrödinger linéaire, dépendant du temps, avec une croissance optimale des normes de Sobolev. La construction est explicite, et s’appuie sur une étude complète de l’équation linéaire de plus bas niveau de Landau avec un potentiel dépendant du temps.

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DOI: 10.5802/ahl.111
Classification: 35Q41, 35B08
Keywords: Linear Schrödinger equation, time-dependent potential, growth of Sobolev norms, reducibility.
Thomann, Laurent 1

1 Institut Élie Cartan, Université de Lorraine, B.P. 70239, F-54506 Vandœuvre-lès-Nancy Cedex, (France)
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     title = {Growth of {Sobolev} norms for linear {Schr\"odinger} operators},
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Thomann, Laurent. Growth of Sobolev norms for linear Schrödinger operators. Annales Henri Lebesgue, Volume 4 (2021), pp. 1595-1618. doi : 10.5802/ahl.111. http://archive.numdam.org/articles/10.5802/ahl.111/

[ABN06] Aftalion, Amandine; Blanc, Xavier; Nier, Francis Lowest Landau level functional and Bargmann spaces for Bose–Einstein condensates, J. Funct. Anal., Volume 241 (2006) no. 2, pp. 661-702 | DOI | MR | Zbl

[ANS19] Arbunich, Jack; Nenciu, Irina; Sparber, Christof Stability and instability properties of rotating Bose–Einstein condensates, Lett. Math. Phys., Volume 109 (2019) no. 6, pp. 1415-1432 | DOI | MR | Zbl

[Bam17] 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

[Bam18] 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

[BBCE17] Biasi, Anxo; Bizoń, Piotr; Craps, Ben; Evnin, Oleg Exact lowest-Landau-level solutions for vortex precession in Bose–Einstein condensates, Phys. Rev. A, Volume 96 (2017) no. 5, 053615 | DOI

[BBE19] Biasi, Anxo; Bizoń, Piotr; Evnin, Oleg Solvable cubic resonant systems, Commun. Math. Phys., Volume 369 (2019) no. 2, pp. 433-456 | DOI | MR | Zbl

[BGMR18] 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 | DOI | MR | Zbl

[BGMR21] Bambusi, Dario; Grébert, Benoît; Maspero, Alberto; Robert, Didier Growth of Sobolev norms for abstract linear Schrödinger equation, J. Eur. Math. Soc. (JEMS), Volume 23 (2021) no. 2, pp. 557-583 | DOI | Zbl

[BM18] Bambusi, Dario; Montalto, Riccardo Reducibility of 1--d Schrödinger equation with time quasiperiodic unbounded perturbations. III, J. Math. Phys., Volume 59 (2018) no. 12, 122702 | Zbl

[Bou99a] Bourgain, Jean Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential, Commun. Math. Phys., Volume 204 (1999) no. 1, pp. 207-247 | DOI | Zbl

[Bou99b] Bourgain, Jean On growth of Sobolev norms in linear Schrödinger equations with smooth time-dependent potential, J. Anal. Math., Volume 77 (1999), pp. 315-348 | DOI | Zbl

[Car91] Carlen, Eric A. Some integral identities and inequalities for entire functions and their application to the coherent state transform, J. Funct. Anal., Volume 97 (1991) no. 1, pp. 231-249 | DOI | MR | Zbl

[Del10] Delort, Jean-Marc Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds, Int. Math. Res. Not., Volume 2010 (2010) no. 12, pp. 2305-2328 | Zbl

[Del14] Delort, Jean-Marc Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential, Commun. Partial Differ. Equations, Volume 39 (2014) no. 1, pp. 1-33 | DOI | Zbl

[EK09] Eliasson, Hakan 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

[Eli01] Eliasson, Hakan 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

[FGH16] Faou, Erwan; Germain, Pierre; Hani, Zaher The weakly nonlinear large box limit of the 2D cubic NLS, J. Am. Math. Soc., Volume 29 (2016) no. 4, pp. 915-982 | DOI | Zbl

[FR20] Faou, Erwan; Raphaël, Pierre On weakly turbulent solutions to the perturbed linear harmonic oscillator (2020) (https://arxiv.org/abs/2006.08206)

[FZ12] Fang, Daoyuan; Zhang, Qidi On growth of Sobolev norms in linear Schrödinger equations with time-dependent Gevrey potential, J. Dyn. Differ. Equations, Volume 24 (2012) no. 2, pp. 151-180 | DOI | Zbl

[GGT19] Gérard, Patrick; Germain, Pierre; Thomann, Laurent On the cubic lowest Landau level equation, Arch. Ration. Mech. Anal., Volume 231 (2019) no. 2, pp. 1073-1128 | DOI | MR | Zbl

[GHT16] Germain, Pierre; Hani, Zaher; Thomann, Laurent On the continuous resonant equation for NLS. I. Deterministic analysis, J. Math. Pures Appl., Volume 105 (2016) no. 1, pp. 131-163 | DOI | MR | Zbl

[GIP09] 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

[GP19] Grébert, Benoît; Paturel, Éric On reducibility of quantum harmonic oscillator on d with quasiperiodic in time potential, Ann. Fac. Sci. Toulouse, Math., Volume 28 (2019) no. 5, pp. 977-1014 | DOI | MR | Zbl

[GPT13] Grébert, Benoît; Paturel, Éric; Thomann, Laurent Beating effects in cubic Schrödinger systems and growth of Sobolev norms, Nonlinearity, Volume 26 (2013) no. 5, pp. 1361-1376 | DOI | Zbl

[GT11] Grébert, Benoît; Thomann, Laurent KAM for the quantum harmonic oscillator, Commun. Math. Phys., Volume 307 (2011) no. 2, pp. 383-427 | DOI | MR | Zbl

[Hel84] Helffer, Bernard Théorie spectrale pour des opérateurs globalement elliptiques, Astérisque, 112, Société Mathématique de France, 1984 | Numdam | Zbl

[HM20] Haus, Emanuele; Maspero, Alberto Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators, J. Funct. Anal., Volume 278 (2020) no. 2, 108316 | MR | Zbl

[LW19] Liang, Zhenguo; Wang, Zhiguo Reducibility of quantum harmonic oscillator on d with differential and quasi-periodic in time potential, J. Differ. Equations, Volume 267 (2019) no. 5, pp. 3355-3395 | DOI | MR | Zbl

[LZZ21] Liang, Zhenguo; Zhao, Zhiyan; Zhou, Qi 1--d quantum harmonic oscillator with time quasi-periodic quadratic perturbation: reducibility and growth of Sobolev norms, J. Math. Pures Appl., Volume 146 (2021), pp. 158-182 | DOI | MR | Zbl

[Mas19] Maspero, Alberto Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations, Math. Res. Lett., Volume 26 (2019) no. 4, pp. 1197-1215 | DOI | MR | Zbl

[MR17] Maspero, Alberto; Robert, Didier On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms, J. Funct. Anal., Volume 273 (2017) no. 2, pp. 721-781 | DOI | Zbl

[Nie07] Nier, Francis Bose–Einstein condensates in the lowest Landau level: Hamiltonian dynamics, Rev. Math. Phys., Volume 19 (2007) no. 1, pp. 101-130 | DOI | MR | Zbl

[Par10] Parmeggiani, Alberto Spectral theory of non-commutative harmonic oscillators: an introduction, Lecture Notes in Mathematics, 1992, Springer, 2010 | DOI | MR | Zbl

[Rob87] Robert, Didier Autour de l’approximation semi-classique, Progress in Mathematics, 68, Birkhäuser, 1987 | Zbl

[ST21] Schwinte, Valentin; Thomann, Laurent Growth of Sobolev norms for coupled Lowest Landau Level equations, Pure Appl. Anal., Volume 3 (2021) no. 1, pp. 189-222 | DOI | MR | Zbl

[Wan08a] Wang, Wei-Min Logarithmic bounds on Sobolev norms for time-dependant linear Schrödinger equations, Commun. Partial Differ. Equations, Volume 33 (2008) no. 12, pp. 2164-2179 | DOI

[Wan08b] 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

[YZ04] Yajima, Kenji; Zhang, Guoping Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity, J. Differ. Equations, Volume 202 (2004) no. 1, pp. 81-110 | DOI | Zbl

[Zhu12] Zhu, Kehe Analysis on Fock spaces, Graduate Texts in Mathematics, 263, Springer, 2012 | Zbl

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