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Wegner Estimates for Sign-Changing Single Site Potentials

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Abstract

We study Anderson and alloy-type random Schrödinger operators on ℓ2(ℤd) and L 2(ℝd). Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. For a certain class of models we prove a Wegner estimate which is linear in the volume of the box and the length of the considered energy interval. The single site potential of the Anderson/alloy-type model does not need to have fixed sign, but it needs be of a generalised step function form. The result implies the Lipschitz continuity of the integrated density of states.

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References

  1. Arveson, W.: A short course on spectral theory. Graduate Texts in Mathematics, vol. 209. Springer, New York (2002)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  4. Bourgain, J.: Anderson localization for quasi-periodic lattice Schrödinger operators on ℤd, d arbitrary. Geom. Funct. Anal. 17(3), 682–706 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bourgain, J.: An approach to Wegner’s estimate using subharmonicity. J. Stat. Phys. 134(5–6), 969–978 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Böttcher, A., Silbermann, B.: Introduction to large truncated Toeplitz matrices. Springer (1999)

  7. Combes, J.-M., Hislop, P.D.: Localization for some continuous, random Hamiltionians in d-dimensions. J. Funct. Anal. 124, 149–180 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  8. Combes, J.-M., Hislop, P.D., Klopp, F.: An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J. 140(3), 469–498 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Combes, J.-M., Hislop, P.D., Klopp, F., Nakamura, S.: The Wegner estimate and the integrated density of states for some random operators. Proc. Indian Acad. Sci. Math. Sci. 112(1), 31–53 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Douglas, R.G., Howe, R.: On the C  ∗ -algebra of Toeplitz operators on the quarter-plane. Trans. Am. Math. Soc. 158, 203–217 (1971)

    MATH  MathSciNet  Google Scholar 

  11. Fischer, W., Hupfer, T., Leschke, H., Müller, P.: Existence of the density of states for multi-dimensional continuum Schrödinger operators with Gaussian random potentials. Commun. Math. Phys. 190(1), 133–141 (1997)

    Article  MATH  ADS  Google Scholar 

  12. Germinet, F., Klein, A.: A characterization of the Anderson metal-insulator transport transition. Duke Math. J. 124(2), 309–350 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Goldstein, M., Schlag, W.: Hölder continuity of the integrated density of states for quasiperiodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. Math. (2), 154(1), 155–203 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hislop, P.D., Klopp, F.: The integrated density of states for some random operators with nonsign definite potentials. J. Funct. Anal. 195(1), 12–47 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge University Press (2004)

  16. Kirsch, W.: Wegner estimates and Anderson localization for alloy-type potentials. Math. Z. 221, 507–512 (1996)

    MATH  MathSciNet  Google Scholar 

  17. Klopp, F.: Localization for some continuous random Schrödinger operators. Commun. Math. Phys. 167, 553–569 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Kirsch, W., Metzger, B.: The integrated density of states for random Schrödinger operators. In: Spectral Theory and Mathematical Physics, pp. 649–698. AMS (2007)

  19. Krüger, H.: Localization for random operators with non-monotone potentials with exponentially decaying correlations. http://arxiv.org/1006.5233

  20. Kozak, A.V., Simonenko, I.B.: Projection methods for solving multidimensional discrete convolution equations. Sib. Mat. Z. 21(2), 119–127, 237 (1980)

    MATH  MathSciNet  Google Scholar 

  21. Kotani, S., Simon, B.: Localization in general one-dimensional random systems. II. Continuum Schrödinger operators. Commun. Math. Phys. 112(1), 103–119 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Kostrykin, V., Veselić, I.: On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials. Math. Z. 252(2), 367–392 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Stollmann, P.: Caught by disorder: bound states in random media. Progress in Mathematical Physics, vol. 20. Birkhäuser (2001)

  24. Tautenhahn, M., Veselić, I.: Spectral properties of discrete alloy-type models. In: XVth International Conference on Mathematical Physics, Prague, 2009. World Scientific (2010)

  25. Veselić, I.: Lipschitz-continuity of the integrated density of states for Gaussian random potentials. (Preprint)

  26. Veselić, I.: Wegner estimate and the density of states of some indefinite alloy type Schrödinger operators. Lett. Math. Phys. 59(3), 199–214 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Veselić, I.: Existence and regularity properties of the integrated density of states of random Schrödinger Operators. Lecture Notes in Mathematics, vol. 1917. Springer, Verlag (2007)

  28. Veselić, I.: Wegner estimate for discrete alloy-type models. Ann. Henri Poincaré. doi:10.1007/s00023-010-0052-5. Earlier version available at http://www.ma.utexas.edu/mp_arc/a/09-100.

  29. Wegner, F.: Bounds on the DOS in disordered systems. Z. Phys. B. 44, 9–15 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  30. Ziemer, W.P.: Weakly Differentiable Functions. Springer, New York (1989)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, 09107, TU-Chemnitz, Germany

    Ivan Veselić

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  1. Ivan Veselić
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This work has been partially supported by the DFG within the Emmy-Noether-Project “Spectral properties of random Schrödinger operators and random operators on manifolds and graphs”.

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Veselić, I. Wegner Estimates for Sign-Changing Single Site Potentials. Math Phys Anal Geom 13, 299–313 (2010). https://doi.org/10.1007/s11040-010-9081-z

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