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Approximate Normal Forms via Floquet–Bloch Theory: Nehorošev Stability for Linear Waves in Quasiperiodic Media

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Abstract

We study the long-time behavior of the Schrödinger flow in a heterogeneous potential \(\lambda V\) with small intensity \(0<\lambda \ll 1\) (or alternatively at high frequencies). The main new ingredient, which we introduce in the general setting of a stationary ergodic potential, is an approximate stationary Floquet–Bloch theory that is used to put the perturbed Schrödinger operator into approximate normal form. We apply this approach to quasiperiodic potentials and establish a Nehorošev-type stability result. In particular, this ensures asymptotic ballistic transport up to a stretched exponential timescale \(\exp (\lambda ^{-\frac{1}{s}})\) for some \(s>0\). More precisely, the approximate normal form leads to an accurate long-time description of the Schrödinger flow as an effective unitary correction of the free flow. The approach is robust and generically applies to linear waves. For classical waves, for instance, this allows to extend diffractive geometric optics to quasiperiodically perturbed media.

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Notes

  1. Similarly as in Lemma 4.1, the factor \(C_\kappa \) could be improved into \(C|\partial B_\kappa |\) if we replace the RHS \(R^{-1}\) in (4.4) by \(R^{-1}|\xi |^{-s_0}\) with \(s_0>d-1\).

References

  1. Allaire, G., Palombaro, M., Rauch, J.: Diffractive behavior of the wave equation in periodic media: weak convergence analysis. Ann. Mat. Pura Appl. (4) 188(4), 561–589 (2009)

    Article  MathSciNet  Google Scholar 

  2. Allaire, G., Palombaro, M., Rauch, J.: Diffractive geometric optics for Bloch wave packets. Arch. Ration. Mech. Anal. 202(2), 373–426 (2011)

    Article  MathSciNet  Google Scholar 

  3. Arnol\(^{\prime }\)d, V.I.: Instability of dynamical systems with many degrees of freedom. Dokl. Akad. Nauk SSSR, 156:9–12 (1964)

  4. Asch, J., Knauf, A.: Motion in periodic potentials. Nonlinearity 11(1), 175–200 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  5. Benoit, A., Duerinckx, M., Gloria, A., Shirley, C.: Approximate spectral theory and wave propagation in quasi-periodic media. In: Séminaire Journées équations aux dérivées partielles, Exp. No. 5 (2017)

  6. Benoit, A., Gloria, A.: Long-time homogenization and asymptotic ballistic transport of classical waves. Ann. Sci. l’ENS 52(3), 703–760 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Bourgain, J., Goldstein, M., Schlag, W.: Anderson localization for Schrödinger operators on \(\mathbb{Z}^{2}\) with quasi-periodic potential. Acta Math. 188(1), 41–86 (2002)

  8. Bourgain, J.: Green’s function estimates for lattice Schrödinger operators and applications. In: Annals of Mathematics Studies, vol. 158, Princeton University Press, Princeton, NJ (2005)

  9. Bourgain, J.: Anderson localization for quasi-periodic lattice Schrödinger operators on \(\mathbb{Z}^d\), \(d\) arbitrary. Geom. Funct. Anal. 17(3), 682–706 (2007)

    Article  MathSciNet  Google Scholar 

  10. Damanik, D.: Schrödinger operators with dynamically defined potentials. Surv. Ergod. Theor. Dyn. Syst. 37, 1681–1764 (2017)

    Article  Google Scholar 

  11. De Giorgi, E.: Sulla convergenza di alcune successioni d’integrali del tipo dell’area. Rend. Mat. 8, 277–294 (1975)

    MathSciNet  MATH  Google Scholar 

  12. Duclos, P., Šťovíček, P., Vittot, M.: Perturbation of an eigenvalue from a dense point spectrum: an example. J. Phys. A 30(20), 7167–7185 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  13. Duerinckx, M., Gloria, A.: Stochastic homogenization of nonconvex unbounded integral functionals with convex growth. Arch. Ration. Mech. Anal. 221(3), 1511–1584 (2016)

    Article  MathSciNet  Google Scholar 

  14. Duerinckx, M., Shirley, C.: A new spectral analysis of stationary random Schrödinger operators. arXiv:2004.12025

  15. Eliasson, L.H.: Floquet solutions for the \(1\)-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146(3), 447–482 (1992)

    Article  ADS  Google Scholar 

  16. Faris, W.G., Lavine, R.B.: Commutators and self-adjointness of Hamiltonian operators. Commun. Math. Phys. 35, 39–48 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  17. Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one-dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys. 132(1), 5–25 (1990)

    Article  ADS  Google Scholar 

  18. Karpeshina, Y., Lee, Y.-R.: Spectral properties of a limit-periodic Schrödinger operator in dimension two. J. Anal. Math. 120, 1–84 (2013)

    Article  MathSciNet  Google Scholar 

  19. Karpeshina, Y., Lee, Y.-R., Shterenberg, R., Stolz, G.: Ballistic transport for the Schrödinger operator with limit-periodic or quasi-periodic potential in dimension two. Commun. Math. Phys. 354(1), 85–113 (2017)

    Article  ADS  Google Scholar 

  20. Karpeshina, Y., Shterenberg, R.: Extended states for the Schrödinger operator with quasi-periodic potential in dimension two. Mem. Am. Math. Soc. 258, 1239 (2019)

    MATH  Google Scholar 

  21. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995)

    Book  Google Scholar 

  22. Kuchment, P.: An overview of periodic elliptic operators. Bull. Am. Math. Soc. (N.S.) 53(3), 343–414 (2016)

    Article  MathSciNet  Google Scholar 

  23. Krueger, H.: Absolutely continuous spectrum for quasi-periodic Schrödinger operators (2013). arXiv:1306.4395

  24. Lin, C.S.: Interpolation inequalities with weights. Commun. Partial Differ. Equ. 11(14), 1515–1538 (1986)

    Article  MathSciNet  Google Scholar 

  25. Nehorošev, N.N.: Action-angle variables, and their generalizations. Trudy Moskov. Mat. Obšč. 26, 181–198 (1972)

    MathSciNet  Google Scholar 

  26. Nehorošev, N.N.: An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Uspehi Mat. Nauk 32(6(198)), 5–66, 287 (1977)

  27. Nehorošev, N.N.: An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II. Trudy Sem. Petrovsk. 5, 5–50 (1979)

    MathSciNet  Google Scholar 

  28. Ozawa, T.: Invariant subspaces for the Schrödinger evolution group. Ann. Inst. H. Poincaré Phys. Théor. 54(1), 43–57 (1991)

    MathSciNet  MATH  Google Scholar 

  29. Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. In: Random fields, vol. I, II (Esztergom, 1979) (volume 27 of Colloq. Math. Soc. János Bolyai, pp. 835–873), North-Holland, Amsterdam (1981)

  30. Reed, M., Simon, B.: Methods of Modern Mathematical Physics: IV—Analysis of Iperators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1978)

  31. Rellich, F.: Perturbation Theory of Eigenvalue Problems. Gordon and Breach Science Publishers, New York (1969)

    MATH  Google Scholar 

  32. Spohn, H.: Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17(6), 385–412 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  33. Trefethen, L.N., Embree, M.: Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005)

    Book  Google Scholar 

  34. Zhao, Z.: Ballistic motion in one-dimensional quasi-periodic discrete Schrödinger equation. Commun. Math. Phys. 347, 511–549 (2016)

    Article  ADS  Google Scholar 

  35. Zhao, Z.: Ballistic transport in one-dimensional quasi-periodic continuous Schrödinger equation. J. Differ. Equ. 262, 4523–4566 (2017)

    Article  ADS  Google Scholar 

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Acknowledgements

The authors warmly thank László Erdős and Tom Spencer for some stimulating discussions on this problem. The work of MD was supported by F.R.S.-FNRS and by the CNRS-Momentum program. Financial support is acknowledged from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM 335410) and European Union’s Horizon 2020 research and innovation programme under grant agreement No 864066.

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Authors and Affiliations

  1. Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris-Saclay, 91400, Orsay, France

    Mitia Duerinckx & Christopher Shirley

  2. Département de Mathématique, Université Libre de Bruxelles, 1050, Brussels, Belgium

    Mitia Duerinckx & Antoine Gloria

  3. Laboratoire Jacques-Louis Lions, CNRS, Sorbonne Université and Institut Universitaire de France (IUF), 75005, Paris, France

    Antoine Gloria

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  1. Mitia Duerinckx
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Correspondence to Antoine Gloria.

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Communicated by W. Schlag.

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Duerinckx, M., Gloria, A. & Shirley, C. Approximate Normal Forms via Floquet–Bloch Theory: Nehorošev Stability for Linear Waves in Quasiperiodic Media. Commun. Math. Phys. 383, 633–683 (2021). https://doi.org/10.1007/s00220-021-03966-7

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