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Communications in Mathematical Physics

, Volume 290, Issue 1, pp 371–387 | Cite as

Growth of Sobolev Norms and Controllability of the Schrödinger Equation

  • Vahagn Nersesyan
Article

Abstract

In this paper we obtain a stabilization result for both linear and nonlinear Schrödinger equations under generic assumptions on the potential. Then we consider the Schrödinger equations with a potential which has a random time-dependent amplitude. We show that if the distribution of the amplitude is sufficiently non-degenerate, then any trajectory of the system is almost surely non-bounded in Sobolev spaces.

Keywords

Function Versus Sobolev Norm Random Potential Unique Continuation Approximate Controllability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agrachev A., Chambrion T.: An estimation of the controllability time for single-input systems on compact Lie groups. J. ESAIM Control Optim. Calc. Var. 12(3), 409–441 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Albert J.H.: Genericity of simple eigenvalues for elliptic PDE’s. Proc. Amer. Math. Soc. 48, 413–418 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Albertini F., D’Alessandro D.: Notions of controllability for bilinear multilevel quantum systems. IEEE Transactions on Automatic Control 48(8), 1399–1403 (2003)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Altafini C.: Controllability of quantum mechanical systems by root space decomposition of su(n). J. Math. Phys. 43(5), 2051–2062 (2002)zbMATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Ball J.M., Marsden J.E., Slemrod M.: Controllability for distributed bilinear systems. SIAM J. Control Optim. 20, 575–597 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Baudouin L., Puel J.-P.: Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Problems 18, 1537–1554 (2001)CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Beauchard K.: Local controllability of a 1-D Schrödinger equation. J. Math. Pures et Appl. 84(7), 851–956 (2005)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Beauchard K., Coron J.-M.: Controllability of a quantum particle in a moving potential well. J. Funct. Anal. 232(2), 328–389 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Beauchard K., Coron J.-M., Mirrahimi M., Rouchon P.: Implicit Lyapunov control of finite dimensional Schrödinger equations. Syst. Cont. Lett. 56, 388–395 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Beauchard, K., Mirrahimi, M.: Approximate stabilization of a quantum particle in a 1D infinite square potential well. http://arxiv.org/abs/0801.1522v1[math.AP], 2008, to apppear SIAMJ Cont. Opt.
  11. 11.
    Bourgain J.: On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential. J. Anal. Math. 77, 315–348 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Burq N.: Contrôle de l’équation des plaques en présence d’obstacles strictement convexes. Mémoire de la S.M.F. 55, 126 (1993)MathSciNetGoogle Scholar
  13. 13.
    Cazenave, T.: Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, 10, Providence, RI: Amer. Math. Soc., 2003Google Scholar
  14. 14.
    Chambrion, T., Mason, P., Sigalotti, M., Boscain, U.: Controllability of the discrete-spectrum Schrödinger equation driven by an external field. http://arxiv.org/abs/0801.4893v3[math.OC], 2008
  15. 15.
    Coron, J.-M.: Control and nonlinearity. Mathematical Surveys and Monographs, Providence, RI: Amer. Math. Soc., 136, 2007Google Scholar
  16. 16.
    Dehman B., Gérard P., Lebeau G.: Stabilization and control for the nonlinear Schrödinger equation on a compact surface. Math. Z. 254(4), 729–749 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Eliasson L.H., Kuksin S.B.: On reducibility of Schrödinger equations with quasiperiodic in time potentials. Commun. Math. Phys. 286(1), 125–135 (2009)CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Erdogan M.B., Killip R., Schlag W.: Energy growth in Schrödinger’s equation with Markovian forcing. Commun. Math. Phys. 240, 1–29 (2003)zbMATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Jerison D., Kenig C.E.: Unique continuation and absence of positive eigenvalues for Schrödinger operators (with an appendix by E. M. Stein). Ann. Math. 121(3), 463–494 (1985)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Lebeau G.: Contrôle de l’équation de Schrödinger. J. Math. Pures Appl. 71, 267–291 (1992)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Machtyngier E., Zuazua E.: Stabilization of the Schrödinger equation. Portugaliae Matematica 51(2), 243–256 (1994)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Mirrahimi, M.: Lyapunov control of a particle in a finite quantum potential well. IEEE Conf. on Decision and Control, San Diego, 2006Google Scholar
  23. 23.
    Nersesyan, V.: Exponential mixing for finite-dimensional approximations of the Schrödinger equation with multiplicative noise. http://arxiv.org/abs/0710.3693v1[math-ph], 2007
  24. 24.
    Nersesyan, V.: Global approximate controllability for Schrödinger equation in higher Sobolev norms. In preparation, 2009Google Scholar
  25. 25.
    Øksendal B.: Stochastic Differential Equations. Springer–Verlag, Berlin-Heidelberg-New York (2003)Google Scholar
  26. 26.
    Pöschel J., Trubowitz E.: Inverse Spectral Theory. Academic Press, New York (1987)zbMATHGoogle Scholar
  27. 27.
    Ramakrishna V., Salapaka M., Dahleh M., Rabitz H., Pierce A.: Controllability of molecular systems. Phys. Rev. A 51(2), 960–966 (1995)CrossRefADSGoogle Scholar
  28. 28.
    Revuz D.: Markov Chains. North–Holland, Amsterdam (1984)zbMATHGoogle Scholar
  29. 29.
    Turinici G., Rabitz H.: Quantum wavefunction controllability. Chem. Phys. 267, 1–9 (2001)CrossRefADSGoogle Scholar
  30. 30.
    Wang W.-M.: Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations. J. Func. Anal. 254(11), 2926–2946 (2008)zbMATHCrossRefGoogle Scholar
  31. 31.
    Wang W.-M.: Logarithmic bounds on Sobolev norms for time dependent linear Schrödinger equations. Commun. PDE 33(12), 2164–2179 (2008)zbMATHCrossRefGoogle Scholar
  32. 32.
    Zuazua E.: Remarks on the controllability of the Schrödinger equation. CRM Proc. Lecture Notes 33, 193–211 (2003)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité de Paris-Sud XIOrsay CedexFrance

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