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Hölder Regularity of Integrated Density of States for the Almost Mathieu Operator in a Perturbative Regime

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Abstract

Consider the Almost Mathieu operator H λ = λ cos 2π(kω +θ)+Δ on the lattice. It is shown that for large λ, the integrated density of states is Hölder continuous of exponent κ < \(\frac{1}{2}\). This result gives a precise version in the perturbative regime of recent work by M. Goldstein and W. Schlag on Hölder regularity of the integrated density of states for 1D quasi-periodic lattice Schrödinger operators, assuming positivity of the Lyapunov exponent (and proven by different means). Our approach provides also a new way to control Green's functions, in the spirit of the author's work in KAM theory. It is by no means restricted to the cosine-potential and extends to band operators.

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References

  1. Aubry, S. and Andre, G.: Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Israel Phys. Soc. 3 (1980), 133–140.

    Google Scholar 

  2. Bellissard, J.: Stability and instability in quantum mechanics, In: Trends and Developments in the Eighties (Bielefeld, 1982/1983), World Scientific, Singapore, 1985, pp. 1–106.

    Google Scholar 

  3. Bourgain. J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math. 148 (1998), 363–439.

    Google Scholar 

  4. Bourgain J.: On Melnikov's persistency problem, Math. Res. Lett. 4 (1997), 445–458.

    Google Scholar 

  5. Bourgain, J. and Goldstein. M.: On nonperturbative localization with quasi-periodic potential, preprint 1999, to appear in Ann. of Math.

  6. Bellissard, J., Lima, R. and Testard, D.: A metal-insulator transition for the almost Mathieu model, Comm. Math. Phys. 88(1983), 207–234.

    Google Scholar 

  7. Bourgain, J. and Schlag, W.: Anderson localization for Schrödinger operators on Ζ with strongly mixing potentials, Preprint 2000, to appear in Comm. Math. Phys.

  8. Craig, W.: Problémes de petits diriseurs dans les équations aux dirivies partielles, Preprint.

  9. Chulaevsky, V. and Sinai. Y.: Anderson localization for multifrequency quasi-periodic potentials in one dimension, Comm. Math. Phys. 125 (1989), 91–121.

    Google Scholar 

  10. Chulaevsky, V. and Spencer, T: Positive Lyapunov exponents for a class of deterministic potentials, Comm. Math. Phys. 168 (1995), 455–466.

    Google Scholar 

  11. Eliasson, L. H.: Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum, Acta Math. 179 (1997), 153–196.

    Google Scholar 

  12. Frölich, J., Spencer, T. and Wittwer, P.: Localization for a class of one-dimensional quasi-periodic operators, Comm. Math. Phys. 132 (1990), 5–25.

    Google Scholar 

  13. Goldstein, M.: Laplace transform method in perturbation theory of the spectrum of Schrödinger operator II (one-dimensional quasi-periodic potentials), Preprint 1992.

  14. Goldstein, M. and Schlag, W.: Holder continuity of integrated density of states for quasi-periodic Schrödinger equation and averages of shifts of subharmonic functions, Preprint 1999.

  15. Herman, M.: Une méthode pour minorer les exposants de Lyapunov et quelques examples montrant le caracteére local d'un théoréme d'Arnold et deMoser sur le tore en dimension 2, Comm. Math. Helv. 58 (1983), 453–502.

    Google Scholar 

  16. Jitomirskaya, S.: Metal-insulator transition for the almost Mathieu Operator, Ann. of Math.

  17. Last, Y.: Almost everything about the almost Mathieu operator I, Proc. XI Internat. Cong. Math. Phys., Paris 1994, Int. Press 1995, pp. 366–372.

  18. Le Page, E.: Regularite du plus grand exposant caractiristique de matrices aleatoires indépendantes et applications, Ann. Inst. Henri Poincaré Ser. B, 25, 109–142.

  19. Last, Y. and Simon, B.: Eigenfunctions, transfermatrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), 329–367.

    Google Scholar 

  20. Peierls, R.: Zur Theorie desDiamagnetismus von Leitungselektronen, Z. Phys., 80 (1933), 763–791.

    Google Scholar 

  21. Sinai, Y.: Anderson localization for one dimensional difference Schrödinger operators with quasi-periodic potential, J. Statist. Phys. 46 (1987), 861–909.

    Google Scholar 

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Authors and Affiliations

  1. Institute for Advanced Studies, Princeton, NJ, 08540, U.S.A.

    J. Bourgain

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Bourgain, J. Hölder Regularity of Integrated Density of States for the Almost Mathieu Operator in a Perturbative Regime. Letters in Mathematical Physics 51, 83–118 (2000). https://doi.org/10.1023/A:1007641323456

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