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Lifshitz Tails for a Class of Schrödinger Operators with Random Breather-Type Potential

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Abstract

We derive bounds on the integrated density of states for a class of Schrödinger operators with a random potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrödinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models, we show that the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This result can be used to prove Anderson localization in certain energy/disorder regimes.

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References

  1. Aizenman M., Elgart A., Naboko S., Schenker J.H., Stolz G.: Moment analysis for localization in random Schrödinger operators. Invent. Math. 163(2), 343–413 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Baker J., Loss M., Stolz G.: Minimizing the ground state energy of an electron in a randomly deformed lattice. Commun. Math. Phys. 283(2), 397–415 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Carmona R., Lacroix J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Combes J.-M., Hislop P.D., Mourre E.: Spectral averaging, perturbation of singular spectra, and localization. Trans. Am. Math. Soc. 348(12), 4883–4894 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Combes J.-M., Hislop P.D., Nakamura S.: The L p-theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random Schrödinger operators. Commun. Math. Phys. 70(218), 113–130 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  7. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Text and Monographs in Physics. Springer, Berlin (1987)

  8. Damanik D., Stollmann P.: Multi-scale analysis implies strong dynamical localization. Geom. Funct. Anal. 11(1), 11–29 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fukushima R.: Brownian survival and Lifshitz tail in perturbed lattice disorder. J. Funct. Anal. 256(9), 2867–2893 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Germinet F., Klein A.: Bootstrap multiscale analysis and localization in random media. Commun. Math. Phys. 222(2), 415–448 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Kirsch, W.: Random Schrödinger operators. In: Holden, H., Jensen, A. (eds.) Schrödinger Operators. Lecture Notes in Physics, vol. 345. Springer, Berlin (1989)

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

    MathSciNet  MATH  Google Scholar 

  14. Kirsch W., Martinelli F.: Large deviations and Lifshitz singularity of the integrated density of states of random Hamitonians. Commun. Math. Phys. 89, 27–40 (1983)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Kirsch, W., Metzger, B.: The integrated density of states for random Schrödinger operators. In: Spectral Theory and Mathematical Physics. Proceedings of Symposia in Pure Mathematics, vol. 76, pp. 649–698. AMS, New York (2007)

  16. Kirsch W., Simon B.: Lifshitz tails for periodic plus random potentials. J. Stat. Phys. 42, 799–808 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Kirsch W., Simon B.: Comparison theorems for the gap of Schrödinger operators. J. Funct. Anal. 75, 396–410 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kirsch W., Stollmann P., Stolz G.: Localization for random perturbations of periodic Schrödinger operators. Random Oper. Stochastic Equations 6(3), 241–268 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kirsch W., Warzel S.: Lifshits tails caused by anisotropic decay: the emergence of a quantum-classical regime. Math. Phys. Anal. Geom. 8(3), 257–285 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Klopp F., Nakamura S.: Spectral extrema and Lifshitz tails for non-monotonous alloy type models. Commun. Math. Phys. 287(3), 1133–1143 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Martinelli F., Holden H.: On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on L 2(R ν). Commun. Math. Phys. 93, 197–217 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Mezincescu G.A.: Lifschitz singularities for periodic operators plus random potentials. J. Stat. Phys. 49(5-6), 1181–1190 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Pastur L.A., Figotin A.L.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992)

    MATH  Google Scholar 

  25. Reed M., Simon B.: Methods of Modern Mathematical Physics IV, Analysis of Operators. Academic Press, San Diego (1978)

    MATH  Google Scholar 

  26. Simon B.: Schrödinger Semigroups. Bull. Am. Math. Soc. 7, 447–526 (1982)

    Article  ADS  MATH  Google Scholar 

  27. Simon B.: Lifschitz tails for the Anderson model. J. Stat. Phys. 38, 65–76 (1985)

    Article  ADS  Google Scholar 

  28. Stollmann P.: Lifshitz asymptotics via linear coupling of disorder. Math. Phys. Anal. Geom. 2(3), 279–289 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Stollmann, P.: Caught by disorder: Bound States in Random Media. Progress in Mathematical Physics, vol. 20. Birkhäuser, Boston (2001)

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

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Authors and Affiliations

  1. Fakultät für Mathematik und Informatik, Fern Universität Hagen, Hagen, Germany

    Werner Kirsch

  2. Fakultät für Mathematik, 09107, Chemnitz, Germany

    Ivan Veselić

Authors
  1. Werner Kirsch
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  2. Ivan Veselić
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Additional information

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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Kirsch, W., Veselić, I. Lifshitz Tails for a Class of Schrödinger Operators with Random Breather-Type Potential. Lett Math Phys 94, 27–39 (2010). https://doi.org/10.1007/s11005-010-0417-1

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  • DOI: https://doi.org/10.1007/s11005-010-0417-1

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