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Long Time Anderson Localization for the Nonlinear Random Schrödinger Equation

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Abstract

We prove long time Anderson localization for the nonlinear random Schrödinger equation for arbitrary ℓ 2 initial data, hence giving an answer to a widely debated question in the physics community. The proof uses a Birkhoff normal form type transform to create a barrier where there is essentially no propagation. One of the new features is that this transform is in a small neighborhood enabling us to treat “rough” data, where there are no moment conditions. The formulation of the present result is inspired by the RAGE theorem.

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References

  1. Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Albanese, C., Fröhlich, J.: Periodic solutions of some infinite-dimensional Hamiltonian systems associated with non-linear partial difference equations I. Commun. Math. Phys. 116, 475–502 (1988)

    Article  MATH  ADS  Google Scholar 

  3. Amrein, W., Georgescu, V.: On the characterization of bound states and scattering states in quantum mechanics. Helv. Phys. Acta 46, 635–658 (1973)

    MathSciNet  Google Scholar 

  4. Bambusi, D., Grébert, B.: Birkhoff normal form for PDE’s with tame modulus. Duke Math. J. 135(3), 507–567 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bourgain, J., Wang, W.-M.: Diffusion bound for a nonlinear Schrödinger equation. In: Mathematical Aspect of Nonlinear Dispersive Equations. Ann. of Math. Stud., pp. 21–42. Princeton University Press, Princeton (2007)

    Google Scholar 

  6. Bourgain, J., Wang, W.-M.: Quasi-periodic solutions of non-linear random Schrödinger equation. J. Eur. Math. Soc. 10, 1–45 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators. Springer, Berlin (1987)

    MATH  Google Scholar 

  8. Enss, V.: Asymptotic completeness for quantum-mechanical potential scattering, I. Short-range potentials. Commun. Math. Phys. 61, 285–291 (1978)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983)

    Article  MATH  ADS  Google Scholar 

  10. Gol’dsheid, Ya., Molchanov, S., Pastur, L.: Pure point spectrum of stochastic one dimensional Schrödinger operators. Funct. Anal. Appl. 11, 1 (1977)

    Article  Google Scholar 

  11. Nersesyan, V.: Growth of Sobolev norms and controllability of Schrödinger equation. arXiv:0804.3982v2 (2008)

  12. Ruelle, D.: A remark on bound states in potential scattering states. Nuovo Cimento 61A, 655–662 (1962)

    ADS  MathSciNet  Google Scholar 

  13. von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124, 285–299 (1989)

    Article  MATH  ADS  Google Scholar 

  14. Wang, W.-M.: Logarithmic bounds on Sobolev norms for time dependent linear Schrödinger equations. Commun. PDE 33, 1–16 (2008)

    Article  Google Scholar 

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Authors and Affiliations

  1. Departement de Mathematique, Universite Paris Sud, 91405, Orsay Cedex, France

    W.-M. Wang & Zhifei Zhang

  2. School of Mathematical Sciences, Peking University, 100871, Beijing, China

    Zhifei Zhang

Authors
  1. W.-M. Wang
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  2. Zhifei Zhang
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Correspondence to W.-M. Wang.

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To J. Fröhlich and T. Spencer on their sixtieth birthdays.

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Wang, WM., Zhang, Z. Long Time Anderson Localization for the Nonlinear Random Schrödinger Equation. J Stat Phys 134, 953–968 (2009). https://doi.org/10.1007/s10955-008-9649-1

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  • DOI: https://doi.org/10.1007/s10955-008-9649-1

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