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Some Abstract Wegner Estimates with Applications

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Abstract

We prove some abstract Wegner bounds for random self-adjoint operators. Applications include elementary proofs of Wegner estimates for discrete and continuous Anderson Hamiltonians with possibly sparse potentials, as well as Wegner bounds for quantum graphs with random edge length or random vertex coupling. We allow the coupling constants describing the randomness to be correlated and to have quite general distributions.

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References

  1. Bogachev, V.I.: Measure theory, vol. I, II. Springer, Berlin (2007)

  2. Boutet de Monvel, A., Chulaevsky, V., Stollmann, P., Suhov, Y.: Wegner-type bounds for a multi-particle continuous Anderson model with an alloy-type external potential. J. Stat. Phys. 138(4-5), 553–566 (2010)

    Google Scholar 

  3. Boutet de Monvel, A., Lenz, D., Stollmann, P.: An uncertainty principle, Wegner estimates and localization near fluctuation boundaries. Math. Z. 269(3–4), 663–670 (2011)

    Google Scholar 

  4. Boutet de Monvel, A., Naboko, S., Stollmann, P., Stolz, G.: Localization near fluctuation boundaries via fractional moments and applications. J. Anal. Math. 100, 83–116 (2006)

    Google Scholar 

  5. Chulaevsky V.: A Wegner-type estimate for correlated potentials. Math. Phys. Anal. Geom. 11(2), 117–129 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chulaevsky V., Suhov Y.: Wegner bounds for a two-particle tight binding model. Commun. Math. Phys. 283, 479–489 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Combes, J-M., Hislop, P., Klopp, F.: Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notes 4, 179–209 (2003)

    Google Scholar 

  8. Combes J-M., Hislop P., 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. Dudley R.M.: Real analysis and probability. In: Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  11. Elgart, A., Klein, A.: Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed anderson models. preprint arXiv: 1301.5268v1 (2013)

  12. Elgart, A., Krüger, H., Tautenhahn, M., Veselić, I.: Discrete Schrödinger operators with random alloy-type potential. Spectral analysis of quantum Hamiltonians, Oper. Theory Adv. Appl., vol. 224, pp. 107–131, Birkhäuser (2012)

  13. Elgart, A., Shamis, M., Sodin, S.: Localisation for non-monotone Schrödinger operators. preprint arXiv: 1201.2211 (2012)

  14. Elgart A., Tautenhahn M., Veselić I.: Anderson localization for a class of models with a sign-indefinite single-site potential via fractional moment method. Ann. Henri Poincaré 128, 1571–1599 (2011)

    Article  ADS  Google Scholar 

  15. Exner P., Helm M., Stollmann P.: Localization on a quantum graph with random potential on the edges. Rev. Math. Phys. 19, 923–939 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Graham B.T., Grimmett G.R.: Influence and sharp-threshold theorems for monotonic measures. Ann. Probab. 34(5), 1726–1745 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hislop P., 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 

  18. Hundertmark D., Killip R., Nakamura S., Stollmann P., Veselić I.: Bounds on the spectral shift function and the density of states. Commun. Math. Phys. 262(2), 489–503 (2006)

    Article  ADS  MATH  Google Scholar 

  19. Kirsch, W.: An invitation to random Schrödinger operators. In: Random Schrödinger operators. Panor. Synthèses, vol. 25, pp. 1–119. Soc. Math. France, Paris (2008)

  20. Kirsch, W.: A Wegner estimate for multi-particle random Hamiltonians. Zh. Mat. Fiz. Anal. Geom. 4(1), 121–127, 203 (2008)

  21. Kirsch W., Martinelli F.: On the ergodic properties of the spectrum of general random operators. J. Reine Angew. Math. 334, 141–156 (1982)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  23. Kirsch W., Veselić I.: Wegner estimate for sparse and other generalized alloy type potentials. Proc. Indian Acad. Sci. Math. Sci. 112(1), 131–146 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kitagaki Y.: Wegner estimates for some random operators with Anderson-type surface potentials. Math. Phys. Anal. Geom. 13(1), 47–67 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kitagaki Y.: Generalized eigenvalue-counting estimates for some random acoustic operators. Kyoto J. Math. 51(2), 439–465 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Klein, A.: Unique continuation principle for spectral projections of Schrödinger operators and optimal Wegner estimates for non-ergodic random Schrödinger operators. Commun. Math. Phys. 323(3), 1229–1246 (2013)

    Google Scholar 

  27. Klein A., Nguyen S.T.: The bootstrap multiscale analysis for the multi-particle Anderson model. J. Stat. Phys. 151(5), 938–973 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Klopp F.: Localization for some continuous random Schrödinger operators. Commun. Math. Phys. 167(3), 553–569 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. Klopp F., Pankrashkin K.: Localization on quantum graphs with random vertex couplings. J. Stat. Phys. 131, 651–673 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. Klopp F., Pankrashkin K.: Localization on quantum graphs with random edge lengths. Lett. Math. Phys. 87, 99–114 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  31. Klopp, F., Zenk, H.: The integrated density of states for an interacting multiparticle homogeneous model and applications to the Anderson model. Adv. Math. Phys. (2009). Art. ID 679827, 15.

  32. Krüger H.: Localization for random operators with non-monotone potentials with exponentially decaying correlations. Ann. Henri Poincaré 13(3), 543–598 (2012)

    Article  ADS  MATH  Google Scholar 

  33. Kuchment P.: Quantum graphs I. Some basic structures. Waves Random Media 14(1), S107–S128 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  34. Kunz H., Souillard B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 78(2), 201–246 (1980)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. Lenz D., Peyerimhoff N., Post O., Veselić I.: Continuity of the integrated density of states on random length metric graphs. Math. Phys. Anal. Geom. 12(3), 219–254 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  36. Peyerimhoff, N., Tautenhahn, M., Veselić, I.: Wegner estimates for alloy-type models with sign-changing exponentially decaying single-site potentials. TU Chemnitz Preprint (2011)

  37. Reed M., Simon B.: Methods of modern mathematical physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978)

    MATH  Google Scholar 

  38. Rojas-Molina, C.: The Anderson model with missing sites. preprint arXiv: 1302.3640 (2013)

  39. Rojas-Molina C., Veselić I.: Scale-free unique continuation estimates and applications to random Schrödinger operators. Commun. Math. Phys. 320(1), 245–274 (2013)

    Article  ADS  MATH  Google Scholar 

  40. Sabri M., Étude de la localisation pour des systèmes désordonnés sur un graphe quantique. PhD Thesis, Univ. Paris Diderot (in preparation)

  41. Simon B.: Lifschitz tails for the Anderson model. J. Stat. Phys. 38(1–2), 65–76 (1985)

    Article  ADS  Google Scholar 

  42. Stollmann P.: Wegner estimates and localization for continuum Anderson models with some singular distributions. Arch. Math. (Basel) 75(4), 307–311 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  43. Stollmann P.: From uncertainty principles to Wegner estimates. Math. Phys. Anal. Geom. 13(2), 145–157 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  44. Veselić I.: Wegner estimate for discrete alloy-type models. Ann. Henri Poincaré 11(5), 991–1005 (2010)

    Article  ADS  MATH  Google Scholar 

  45. Veselić I.: Wegner estimates for sign-changing single site potentials. Math. Phys. Anal. Geom. 13(4), 299–313 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  46. Wegner F.: Bounds on the density of states in disordered systems. Z. Phys. B 44(1-2), 9–15 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  47. Weidmann J.: Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68. Springer, New York (1980)

    Book  Google Scholar 

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, Université Paris Diderot Paris 7, Bâtiment Sophie Germain, 75013, Paris, France

    Mostafa Sabri

  2. Department of Mathematics, Faculty of Science, Cairo University, Cairo, 12613, Egypt

    Mostafa Sabri

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Sabri, M. Some Abstract Wegner Estimates with Applications. Lett Math Phys 104, 311–339 (2014). https://doi.org/10.1007/s11005-013-0666-x

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  • DOI: https://doi.org/10.1007/s11005-013-0666-x

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