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Resonance Tongues and Instability Pockets in the Quasi–Periodic Hill–Schrödinger Equation

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This paper concerns Hill’s equation with a (parametric) forcing that is real analytic and quasi-periodic with frequency vector ωℝd and a ‘frequency’ (or ‘energy’) parameter a and a small parameter b. The 1-dimensional Schrödinger equation with quasi-periodic potential occurs as a particular case. In the parameter plane ℝ2={a, b}, for small values of b we show the following. The resonance ‘‘tongues’’ with rotation number \({{\frac{{1}}{{2}}\langle{{\bf{ k}}},\omega\rangle,{{\bf{ k}}}\in\mathbb{{Z}}^d}}\) have C -boundary curves. Our arguments are based on reducibility and certain properties of the Schrödinger operator with quasi-periodic potential. Analogous to the case of Hill’s equation with periodic forcing (i.e., d=1), several further results are obtained with respect to the geometry of the tongues. One result regards transversality of the boundaries at b=0. Another result concerns the generic occurrence of instability pockets in the tongues in a reversible near-Mathieu case, that may depend on several deformation parameters. These pockets describe the generic opening and closing behaviour of spectral gaps of the Schrödinger operator in dependence of the parameter b. This result uses a refined averaging technique. Also consequences are given for the behaviour of the Lyapunov exponent and rotation number in dependence of a for fixed b.

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References

  1. Arnol’d, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Berlin-Heidelberg-New York: Springer-Verlag, 1983

  2. Arnol’d, V.I.: Remarks on perturbation theory for problems of Mathieu type. Uspekhi Mat. Nauk 38(4(232)), 189–203 (1983)

    MATH  Google Scholar 

  3. Arnol’d, V.I.: Mathematical methods of classical mechanics. Berlin-Heidelberg-New York: Springer-Verlag, 1996

  4. Arnol’d, V.I., (ed.): Dynamical Systems V: Bifurcation Theory and Catastrophe Theory, Volume 5 of Encyclopædia of Mathematical Sciences. Berlin-Heidelberg-New York: Springer-Verlag, 1994

  5. Aubry, S., André, G.: Analyticity breaking and Anderson localization in incommensurate lattices. In: Group theoretical methods in physics, (Proc. Eighth Internat. Colloq., Kiryat Anavim, 1979), Bristol: Hilger, 1980, pp. 133–164

  6. Broer, H.W., Hoveijn, I., van Noort, M., Vegter, G.: The inverted pendulum: A singularity theory approach. J. Diff. Eqs. 157(1), 120–149 (1999)

    MATH  Google Scholar 

  7. Broer, H.W., Huitema, G.B., Sevryuk, M.B.: Quasi-periodic motions in families of dynamical systems, order amidst chaos. Lecture Notes in Math. 1645 Berlin: Springer-Verlag, 1996

  8. Broer, H.W., Vegter, G.: Bifurcational aspects of parametric resonance. In: Dynamics reported: Expositions in dynamical systems, Berlin: Springer, 1992, pp. 1–53

  9. Broer, H.W., Hoveijn, I., Lunter, G.A., Vegter, G.: Resonances in a spring-pendulum: Algorithms for equivariant singularity theory. Nonlinearity 11, 1–37 (1998)

    Article  Google Scholar 

  10. Broer, H.W., Hoveijn, I., Lunter, G.A., Vegter, G.: Bifurcations in Hamiltonian Systems, computing singularities by Gröbner bases. Lecture Notes in Math. 1806 Berlin: Springer-Verlag, 2003

  11. Broer, H.W., Hoveijn, I., van Noort, M.: A reversible bifurcation analysis of the inverted pendulum. Physica D 112, 50–63 (1997)

    Google Scholar 

  12. Broer, H.W., Levi, M.: Geometrical aspects of stability theory for Hill’s equations. Arch. Ration. Mech. Anal. 131(3), 225–240 (1995)

    MATH  Google Scholar 

  13. Broer, H.W., Levi, M., Simó, C.: Geometry of stability zones of Hill’s equations. In progress, 2003

  14. Broer, H.W., Lunter, G.A., Vegter, G.: Equivariant singularity theory with distinguished parameters: Two case-studies of resonant Hamiltonian systems. Physica D 112, 64–80 (1997)

    Google Scholar 

  15. Broer, H.W., Simó, C.: Hill’s equation with quasi-periodic forcing: Resonance tongues, instability pockets and global phenomena. Bol. Soc. Brasil. Mat. (N.S.) 29(2), 253–293 (1998)

    MATH  Google Scholar 

  16. Broer, H.W., Simó, C.: Resonance tongues in Hill’s equations: A geometric approach. J. Diff. Eqs. 166(2), 290–327 (2000)

    MATH  Google Scholar 

  17. Broer, H.W., Golubitsky, M., Vegter, G.: The geometry of resonance tongues: A Singularity Theory approach. Nonlinearity 16, 1511–1538 (2003)

    Google Scholar 

  18. Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. Probability and its Applications. Basel-Boston: Birkhäuser, 1990

  19. Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York-Toronto-London: McGraw-Hill Book Company, Inc., 1955

  20. De Concini, C., Johnson, R.A.: The algebraic-geometric AKNS potentials. Ergodic Theory Dynam. Systems 7(1), 1–24 (1987)

    MATH  Google Scholar 

  21. Dinaburg, E.I., Sinai, Y.G.: The one-dimensional Schrödinger equation with quasi-periodic potential. Funkt. Anal. i. Priloz. 9, 8–21 (1975)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. Fabbri, R., Johnson, R., Pavani, R.: On the nature of the spectrum of the quasi-periodic Schrödinger operator. Nonlinear Anal. Real World Appl. 3(1), 37–59 (2002)

    Article  MATH  Google Scholar 

  24. 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)

    Google Scholar 

  25. Fulton, W.: Algebraic curves. An introduction to algebraic geometry. New York-Amsterdam: W. A. Benjamin, Inc., 1969

  26. Giorgilli, A., Galgani, L.: Formal integrals for an autonomous Hamiltonian system near an equilibrium point. Celestial Mech. 17(3), 267–280 (1978)

    MATH  Google Scholar 

  27. Hale, J.K.: Oscillations in nonlinear systems. New York: Dover Publications Inc., 1992. Corrected reprint of the 1963 original

  28. Hochstadt, H.: The functions of mathematical physics. New York: Dover Publications Inc., Second edition, 1986

  29. Johnson, R.: Cantor spectrum for the quasi-periodic Schrödinger equation. J. Diff. Eq. 91, 88–110 (1991)

    MathSciNet  MATH  Google Scholar 

  30. Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84, 403–438 (1982)

    MathSciNet  MATH  Google Scholar 

  31. Levy, D.M., Keller, J.B.: Instability intervals of Hill’s equation. Commun. Pure Appl. Math. 16, 469–479 (1963)

    MATH  Google Scholar 

  32. Magnus, W., Winkler, S.: Hill’s equation. New York: Dover Publications Inc., 1979. Corrected reprint of the 1966 edition

  33. Moser, J.: An example of Schrödinger equation with almost periodic potential and nowhere dense spectrum. Comment. Math. Helvetici 56, 198–224 (1981)

    MathSciNet  MATH  Google Scholar 

  34. Moser, J., Pöschel, J.: An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helvetici 59, 39–85 (1984)

    MathSciNet  MATH  Google Scholar 

  35. Núñez, C.: Extension of a Moser-Pöschel theorem for the Schrödinger equation with ergodic potential. In: XIV CEDYA/IV Congress of Applied Mathematics (Spanish)(Vic, 1995), (electronic). Barcelona: Univ. Barcelona, 1995, 10 pp.

    Google Scholar 

  36. Obaya, R., Paramio, M.: Directional differentiability of the rotation number for the almost periodic Schrödinger equation. Duke Math. J. 66, 521–552 (1992)

    MathSciNet  MATH  Google Scholar 

  37. Olvera, A., Simó, C.: Normal forms close to invariant circles of twist maps. In: C. Alsina, J. Llibre, Ch. Mira, C. Simó, G.Targonski, and R. Thibault, (eds.), European Conference on Iteration Theory (ECIT 87). Singapore: World Scientific, 1989, pp. 438–443

  38. Olvera, A., Simó, C.: Normal forms of twist maps in a resonant annulus. Preprint, 1998

  39. Oxtoby, J.: Measure and Category. Berlin: Springer-Verlag, 1971

  40. Pastur, L., Figotin, A.: Spectra of random and almost-periodic operators. Berlin: Springer-Verlag, 1992

  41. Puig, J.: Reducibility of linear differential equations with quasi-periodic coefficients: A survey. Preprint University of Barcelona, 2003

  42. Puig, J., Simó, C.: Analytic families of reducible linear quasi-periodic equations. In progress, 2003

  43. Sacker, R.J., Sell, G.: A spectral theory for linear differential systems. J. Diff. Eq. 27, 320–358 (1978)

    MATH  Google Scholar 

  44. Simó, C.: Averaging under fast quasiperiodic forcing. In: Hamiltonian mechanics (Toruń, 1993). New York: Plenum, 1994, pp. 13–34

  45. Simon, B.: Almost periodic Schrödinger operators: A review. Adv. in Appl. Math. 3(4), 463–490 (1982)

    MATH  Google Scholar 

  46. Stoker, J.J.: Nonlinear vibrations in mechanical and electrical systems. Wiley Classics Library. New York: John Wiley & Sons Inc., 1992. Reprint of the 1950 original, A Wiley-Interscience Publication

  47. Van der Pol, B., Strutt, M.J.O.: On the stability of the solutions of Mathieu’s equation. The London, Edinburgh and Dublin Phil. Mag. 7th series 5, 18–38 (1928)

    Google Scholar 

  48. Whittaker, E.T., Watson, G.N.: A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: With an account of the principal transcendental functions. New York: Cambridge University Press, 1962

  49. Yakubovich, V.A., Starzhinskii, V.M.: Linear differential equations with periodic coefficients. 1, 2. New York-Toronto, Ont.: Halsted Press [John Wiley & Sons], 1975

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Communicated by G. Gallavotti

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Broer, H., Puig, J. & Simó, C. Resonance Tongues and Instability Pockets in the Quasi–Periodic Hill–Schrödinger Equation. Commun. Math. Phys. 241, 467–503 (2003). https://doi.org/10.1007/s00220-003-0935-0

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