Abstract
We study the Schrödinger equation on \({\mathbb{R}}\) with a potential behaving as \({x^{2l}}\) at infinity, \({l \in [1, + \infty)}\) and with a small time quasiperiodic perturbation. We prove that if the perturbation belongs to a class of unbounded symbols including smooth potentials and magnetic type terms with controlled growth at infinity, then the system is reducible.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bambusi D.: Long time stability of some small amplitude solutions in nonlinear Schrödinger equations. Commun. Math. Phys. 189(1), 205–226 (1997)
Bambusi, D.: Reducibility of 1-D Schrödinger equation with time quasiperiodic unbounded perturbations, I (2016). arXiv:1606.04494 [math.DS]
Baldi P., Berti M., Montalto R.: KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation. Math. Ann. 359(1–2), 471–536 (2014)
Bambusi D., Giorgilli A.: Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems. J. Stat. Phys. 71(3–4), 569–606 (1993)
Bambusi D., Graffi S.: Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods. Commun. Math. Phys. 219(2), 465–480 (2001)
Berti, M., Montalto, R.: Quasi-periodic standing wave solutions of gravity-capillary water waves (2016). arXiv:1602.02411 [math.AP]
Combescure M.: The quantum stability problem for time-periodic perturbations of the harmonic oscillator. Ann. Inst. H. Poincaré Phys. Théor. 47(1), 63–83 (1987)
Delort J.-M.: Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential. Commun. Partial Differ. Equ. 39(1), 1–33 (2014)
Duclos P., Lev O., Šťovíček P., Vittot M.: Weakly regular Floquet Hamiltonians with pure point spectrum. Rev. Math. Phys. 14(6), 531–568 (2002)
Duclos P., Šťovíček P.: Floquet Hamiltonians with pure point spectrum. Commun. Math. Phys. 177(2), 327–347 (1996)
Eliasson H.L., Kuksin S.B.: On reducibility of Schrödinger equations with quasiperiodic in time potentials. Commun. Math. Phys. 286(1), 125–135 (2009)
Feola R., Procesi M.: Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations. J. Differ. Equ. 259(7), 3389–3447 (2015)
Grébert, B., Paturel, E.: On reducibility of quantum harmonic oscillator on \({{\mathbb{R}}^d}\) with quasiperiodic in time potential (2016). arXiv:1603.07455 [math.AP]
Grébert B., Thomann L.: KAM for the quantum harmonic oscillator. Commun. Math. Phys. 307(2), 383–427 (2011)
Graffi S., Yajima K.: Absolute continuity of the Floquet spectrum for a nonlinearly forced harmonic oscillator. Commun. Math. Phys. 215(2), 245–250 (2000)
Helffer B., Robert D.: Asymptotique des niveaux d’énergie pour des hamiltoniens à un degré de liberté. Duke Math. J. 49(4), 853–868 (1982)
Helffer B., Robert D.: Propriétés asymptotiques du spectre d’opérateurs pseudodifférentiels sur r n. Commun. Partial Differ. Equ. 7(7), 795–882 (1982)
Kuksin, S.B.: Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum. Funktsional. Anal. Prilozhen. 21(3), 22–37, 95 (1987)
Kuksin, S.B.: Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Volume 1556 of Lecture Notes in Mathematics. Springer, Berlin (1993)
Kuksin S.B.: On small-denominators equations with large variable coefficients. Z. Angew Math. Phys. 48(2), 262–271 (1997)
Liu J., Yuan X.: Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient. Commun. Pure Appl. Math. 63(9), 1145–1172 (2010)
Montalto, R.: KAM for quasi-linear and fully nonlinear perturbations of Airy and KdV equations. Ph.D. Thesis, SISSA—ISAS (2014)
Maspero, A., Robert, D.: On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms (2016) (Preprint)
I. Plotnikov, J.F. Toland: Nash-Moser theory for standing water waves. Arch. Ration Mech. Anal. 159(1), 1–83 (2001)
Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970)
Wang W.-M.: Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations. Commun. Math. Phys. 277(2), 459–496 (2008)
Wayne C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127(3), 479–528 (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by W. Schlag
Rights and permissions
About this article
Cite this article
Bambusi, D. Reducibility of 1-d Schrödinger Equation with Time Quasiperiodic Unbounded Perturbations, II. Commun. Math. Phys. 353, 353–378 (2017). https://doi.org/10.1007/s00220-016-2825-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-016-2825-2