KAM for the Quantum Harmonic Oscillator

  • Benoît GrébertEmail author
  • Laurent Thomann


In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. Pöschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schrödinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schrödinger equations with the harmonic potential and a quasi periodic in time potential.


Normal Form Harmonic Oscillator Harmonic Potential Sobolev Embedding Hermite Function 
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  1. 1.
    Bambusi D., Graffi S.: Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM method. Commun. Math. Phys. 219(2), 465–480 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. Arch. Rat. Mech. Anal. 82(4), 313–345 and 347–375 (1983)Google Scholar
  3. 3.
    Carles R.: Rotating points for the conformal NLS scattering operator. Dyn. Part. Diff. Eq. 6(1), 35–51 (2009)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Delort, J.-M.: Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential. Preprint Google Scholar
  5. 5.
    Eliasson, L.H.: Almost reducibility of linear quasi-periodic systems. In available at, 2010 Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Providence, RI: Amer. Math. Soc., 2001, pp. 679–705
  6. 6.
    Eliasson L.H., Kuksin S.B.: KAM for the nonlinear Schrödinger equation. Ann. of Math. (2) 172(1), 371–435 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Eliasson L.H., Kuksin S.B.: On reducibility of Schrödinger equations with quasiperiodic in time potentials. Commun. Math. Phys. 286(1), 125–135 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Enss V., Veselic K.: Bound states and propagating states for time-dependent hamiltonians. Ann. IHP 39(2), 159–191 (1983)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Grébert B., Imekraz R., Paturel É.: Normal forms for semilinear quantum harmonic oscillators. Commun. Math. Phys. 291, 763–798 (2009)ADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Kuksin S.B.: Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Funct. Anal. Appl. 21, 192–205 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kuksin, S.B.: Nearly integrable infinite-dimensional Hamiltonian systems. Lecture Notes in Mathematics, 1556. Berlin: Springer-Verlag, 1993Google Scholar
  12. 12.
    Kuksin S.B.: Analysis of Hamiltonian PDEs Oxford Lecture Series in Mathematics and its Applications, 19. Oxford University Press, Oxford (2000)Google Scholar
  13. 13.
    S.B. , S.B. , S.B. , S.B. : Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. of Math. 143, 149–179 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Liu J., Yuan X.: Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient. Comm. Pure Appl. Math. 63(9), 1145–1172 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Pitaevski L.P., Stringari S.: Bose-Einstein Condensation. Oxford University Press, Oxford (2003)Google Scholar
  16. 16.
    Pöschel J.: A KAM-theorem for some nonlinear partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23(1), 119–148 (1996)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Pöschel J.: On elliptic lower-dimensional tori in Hamiltonian systems. Math. Z. 202(4), 559–608 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Pöschel J.: Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helv. 71, 269–296 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Reed M., Simon B.: Methods of modern mathematical physics IV Analysis of operators. Academic Press, New York (1978)zbMATHGoogle Scholar
  20. 20.
    Struwe M.: Variational methods Applications to nonlinear partial differential equations and Hamiltonian systems Fourth edition. Springer-Verlag, Berlin (2008)Google Scholar
  21. 21.
    Yajima K., Kitada H.: Bound states and scattering states for time periodic Hamiltonians. Ann. Inst. H. Poincaré Sect. A (N.S.) 39(2), 145–157 (1983)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Yajima K., Zhang G.: Smoothing property for Schrödinger equations with potential superquadratic at infinity. Commun. Math. Phys. 221(3), 573–590 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Wang W.M.: Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations. Commun. Math. Phys. 277, 459–496 (2008)ADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques J. LerayUniversité de Nantes, UMR CNRS 6629Nantes Cedex 03France

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