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Regular and Chaotic Dynamics

, Volume 16, Issue 1–2, pp 61–78 | Cite as

Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies

  • Joaquim Puig
  • Carles Simó
Article

Abstract

In this paper we investigate numerically the following Hill’s equation x″ + (a + bq(t))x = 0 where \( q(t) = \cos t + \cos \sqrt {2t} + \cos \sqrt {3t} \) is a quasi-periodic forcing with three rationally independent frequencies. It appears, also, as the eigenvalue equation of a Schrödinger operator with quasi-periodic potential.

Massive numerical computations were performed for the rotation number and the Lyapunov exponent in order to detect open and collapsed gaps, resonance tongues. Our results show that the quasi-periodic case with three independent frequencies is very different not only from the periodic analogs, but also from the case of two frequencies. Indeed, for large values of b the spectrum contains open intervals at the bottom. From a dynamical point of view we numerically give evidence of the existence of open intervals of a, for large b, where the system is nonuniformly hyperbolic: the system does not have an exponential dichotomy but the Lyapunov exponent is positive. In contrast with the region with zero Lyapunov exponents, both the rotation number and the Lyapunov exponent do not seem to have square root behavior at endpoints of gaps. The rate of convergence to the rotation number and the Lyapunov exponent in the nonuniformly hyperbolic case is also seen to be different from the reducible case.

Keywords

quasi-periodic Schrödinger operators quasi-periodic cocycles and skew-products spectral gaps resonance tongues rotation number Lyapunov exponent numerical explorations 

MSC2010 numbers

37B55 35J10 

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References

  1. 1.
    Moser, J. and Pöschel, J., An Extension of a Result by Dinaburg and Sinai on Quasi-periodic Potentials, Comment. Math. Helvetici, 1984, vol. 59, pp. 39–85.MATHCrossRefGoogle Scholar
  2. 2.
    Eliasson, L.H., Floquet Solutions for the One-dimensional Quasi-periodic Schrödinger Equation, Comm. Math. Phys., 1992, vol. 146, pp. 447–482.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Broer, H.W., Puig, J., and Simó, C., Resonance Tongues and Instability Pockets in the Quasi-periodic Hill-Schrödinger Equation, Comm. Math. Phys, 2003, vol. 241, pp. 467–503.MATHMathSciNetGoogle Scholar
  4. 4.
    Puig, J. and Simó, C., Analytic Families of Reducible Linear Quasi-periodic Differential Equations, Ergodic Theory Dynam. Systems, 2006, vol. 26, no. 2, pp. 481–524.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Magnus, W. and Winkler, S., Hill’s Equation, New York: Dover Publications Inc., 1979. Corrected reprint of the 1966 edition.Google Scholar
  6. 6.
    Broer, H.W. and Simó, C., Resonance Tongues in Hill’s Equations: a Geometric Approach, J. Differential Equations, 2000, vol. 166, no. 2, pp. 290–327.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    De Concini, C. and Johnson, R.A., The Algebraic-Geometric AKNS Potentials, Ergodic Theory Dynam. Systems, 1987, vol. 7, no. 1, pp. 1–24.MATHMathSciNetGoogle Scholar
  8. 8.
    Johnson, R.A., Cantor Spectrum for the Quasi-periodic Schrödinger Equation, J. Diff. Eq., 1991, vol. 91, pp. 88–110.MATHCrossRefGoogle Scholar
  9. 9.
    Fabbri, R., Johnson, R.A., and Pavani, R., On the Nature of the Spectrum of the Quasi-periodic Schrödinger Operator, Nonlinear Anal. Real World Appl., 2002, vol. 3, no. 1, pp. 37–59.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cong, N.D. and Fabbri, R., On the Spectrum of the One-dimensional Schrödinger Operator, Discrete Contin. Dyn. Syst., Ser. B, 2008, vol. 9, nos 3–4, pp. 541–554.MATHMathSciNetGoogle Scholar
  11. 11.
    Avila, A., Bochi, J., and Damanik, D., Cantor Spectrum for Schrödinger Operators with Potentials Arising from Generalized Skew-shifts, Duke Math. J., 2009, vol. 146, no. 2, pp. 253–280.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Broer, H.W. and Simó, C., Hill’s Equation with Quasi-periodic Forcing: Resonance Tongues, Instability Pockets and Global Phenomena, Bol. Soc. Brasil. Mat. (N.S.), 1998, vol. 29, no. 2, pp. 253–293.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bogoljubov, N.N., Mitropoliskii, J.A., and Samoĭlenko, A.M., Methods of Accelerated Convergence in Nonlinear Mechanics, Delhi: Hindustan Publishing Corp., 1976.Google Scholar
  14. 14.
    Jorba, À. and Simó, C., On the Reducibility of Linear Differential Equations with Quasiperiodic Coefficients, J. Differential Equations, 1992, vol. 98, no. 1, pp. 111–124.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    He, H.L. and You, J., Full Measure Reducibility for Generic One-parameter Family of Quasi-periodic Linear Systems. Journal of Dynamics and Differential Equations, 2008, vol. 20, pp. 1–36.CrossRefMathSciNetGoogle Scholar
  16. 16.
    Sacker, R.J. and Sell, G.R., A Spectral Theory for Linear Differential Systems, J. Diff. Eq., 1978, vol. 27, pp. 320–358.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kingman, J.F.C., The Ergodic Theory of Subadditive Stochastic Processes, J. Roy. Statist. Soc. Ser. B, 1968, vol. 30, pp. 499–510.MATHMathSciNetGoogle Scholar
  18. 18.
    Johnson, R.A. and Moser, J., The Rotation Number for Almost Periodic Potentials, Commun. Math. Phys., 1982, vol. 84, pp. 403–438.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Herman, M.R., Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helvetici, 1983, vol. 58, no. 3, pp. 453–502.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Broer, H.W. and Levi, M., Geometrical Aspects of Stability Theory for Hill’s Equations, Arch. Rational Mech. Anal., 1995, vol. 131, no. 3, pp. 225–240.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Puig, J. and Simó, C., Resonance Tongues and Spectral Gaps in Quasi-periodic Schrödinger Operators with One or More Frequencies. A Numerical Exploration, J. Dynam. Differential Equations, 2010 (to appear).Google Scholar
  22. 22.
    Hadj Amor, S., Hölder Continuity of the Rotation Number for Quasi-periodic Co-cycles in SL(2,R), Comm. Math. Phys., 2009, vol. 287, no. 2, pp. 565–588.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Dinaburg, E.I. and Sinai, Y.G., The One-dimensional Schrödinger Equation with Quasi-periodic Potential, Funkt. Anal. i Prilozh., 1975, vol. 9, pp. 8–21 (Russian).CrossRefMathSciNetGoogle Scholar
  24. 24.
    Simó, C., Averaging Under Fast Quasiperiodic Forcing, in Hamiltonian mechanics (Toruń, 1993), New York: Plenum, 1994, pp. 13–34.Google Scholar
  25. 25.
    Delyon, F. and Foulon, P., Adiabatic Theory, Liapunov Exponents, and Rotation Number for Quadratic Hamiltonians, J. Stat. Phys., 1987, vol. 49, nos 3–4, pp. 829–840.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Delyon, F. and Foulon, P., Adiabatic Invariants and Asymptotic Behavior of Lyapunov Exponents of the Schrödinger Equation, J. Statist. Phys., 1986, vol. 45, nos 1–2, pp. 41–47.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Puig, J., Cantor Spectrum and KDS Eigenstates, Comm. Math. Phys., 2006, vol. 267, no. 3, pp. 735–740.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Fröhlich, J., Spencer, T., and Wittwer, P., Localization for a Class of One-dimensional Quasi-periodic Schrödinger Operators, Comm. Math. Phys., 1990, vol. 132, no. 1, pp. 5–25.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Bjerklöv, K., Positive Lyapunov Exponents for Continuous Quasiperiodic Schrödinger Equations, J. Math. Phys., 2006, vol. 47, 022702, 4 pages.CrossRefMathSciNetGoogle Scholar
  30. 30.
    Jitomirskaya, S., Ergodic Schrödinger Operators (On One Foot), in Spectral Theory and Mathematical Physics. A Festschrift in Honor of Barry Simon’s 60th birthday, Proc. Sympos. Pure Math., vol. 76, Part 2, Providence, RI: AMS, 2007, pp. 613–647.Google Scholar
  31. 31.
    Sinai, Ya.G., Anderson Localization for One-dimensional Difference Schrödinger Operator with Quasiperiodic Potential, J. Statist. Phys., 1987, vol. 46, nos 5–6, pp. 861–909.CrossRefMathSciNetGoogle Scholar
  32. 32.
    Goldstein, M. and Schlag, W., Fine Properties of the Integrated Density of States and a Quantitative Separation Property of the Dirichlet Eigenvalues, Geom. Funct. Anal., 2008, vol. 18, no. 3, pp. 755–869.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Haro, À. and Puig, J., Aubry Duality for Quasi-periodic Potentials I: Lyapunov Exponents and Cantor Spectrum (in preparation).Google Scholar
  34. 34.
    Chulaevsky, V.A. and Sinaĭ, Ya.G., Anderson Localization for the 1-D Discrete Schrödinger Operator with Two-frequency Potential, Comm. Math. Phys., 1989, vol. 125, no. 1, pp. 91–112.CrossRefMathSciNetGoogle Scholar
  35. 35.
    Jorba, À. and Zou, M., A Software Package for the Numerical Integration of ODEs by Means of Highorder Taylor Methods, Experimental Mathematics, 2005, vol. 14, no. 1, pp. 99–117.MATHMathSciNetGoogle Scholar
  36. 36.
    Simó, C., Taylor Method for the Integration of ODE. Lectures given at the Advanced School on Long Time Integrations, 2007, http://www.maia.ub.es/dsg/2007/0708simo.ps.gz
  37. 37.
    Cincotta, P.M., Giordano, C.M., and Simó, C., Phase Space Structure of Multi-dimensional Systems by Means of the Mean Exponential Growth Factor of Nearby Orbits, Phys. D, 2003, vol. 182, nos 3–4, pp. 151–178.MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Avila, A. and Krikorian, R., Reducibility or Non-uniform Hyperbolicity for Quasiperiodic Schrödinger Cocycles, Ann. of Math., 2006, vol. 164, no. 3, pp. 249–294.MathSciNetGoogle Scholar
  39. 39.
    Deift, P. and Simon, B., Almost Periodic Schrödinger Operators III. The Absolute Continuous Spectrum, Comm. Math. Phys., 1983, vol. 90, pp. 389–341.MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Bjerklov, K. and Jager, T., Rotation Numbers for Quasiperiodically Forced Circle Maps-Mode-Locking vs. Strict Monotonicity, J. Am. Math. Soc., 2009, vol. 22, no. 2, pp. 353–362.CrossRefMathSciNetGoogle Scholar
  41. 41.
    Nowak, W.G., On Simultaneous Diophantine Approximation, Rendiconti del Circolo Matematico di Palermo, 1984, vol. 33, no. 3, pp. 456–460.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

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