Letters in Mathematical Physics

, Volume 104, Issue 3, pp 311–339 | Cite as

Some Abstract Wegner Estimates with Applications

  • Mostafa Sabri


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.

Mathematics Subject Classification (1991)

Primary 82B44 Secondary 47B80 34B45 


Wegner estimates random operators sparse potentials quantum graphs 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bogachev, V.I.: Measure theory, vol. I, II. Springer, Berlin (2007)Google Scholar
  2. 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. 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. 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. 5.
    Chulaevsky V.: A Wegner-type estimate for correlated potentials. Math. Phys. Anal. Geom. 11(2), 117–129 (2008)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Chulaevsky V., Suhov Y.: Wegner bounds for a two-particle tight binding model. Commun. Math. Phys. 283, 479–489 (2008)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 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. 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)CrossRefMATHMathSciNetGoogle Scholar
  9. 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)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Dudley R.M.: Real analysis and probability. In: Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  11. 11.
    Elgart, A., Klein, A.: Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed anderson models. preprint arXiv: 1301.5268v1 (2013)Google Scholar
  12. 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)Google Scholar
  13. 13.
    Elgart, A., Shamis, M., Sodin, S.: Localisation for non-monotone Schrödinger operators. preprint arXiv: 1201.2211 (2012)Google Scholar
  14. 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)ADSCrossRefGoogle Scholar
  15. 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)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Graham B.T., Grimmett G.R.: Influence and sharp-threshold theorems for monotonic measures. Ann. Probab. 34(5), 1726–1745 (2006)CrossRefMATHMathSciNetGoogle Scholar
  17. 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)CrossRefMATHMathSciNetGoogle Scholar
  18. 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)ADSCrossRefMATHGoogle Scholar
  19. 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)Google Scholar
  20. 20.
    Kirsch, W.: A Wegner estimate for multi-particle random Hamiltonians. Zh. Mat. Fiz. Anal. Geom. 4(1), 121–127, 203 (2008)Google Scholar
  21. 21.
    Kirsch W., Martinelli F.: On the ergodic properties of the spectrum of general random operators. J. Reine Angew. Math. 334, 141–156 (1982)MATHMathSciNetGoogle Scholar
  22. 22.
    Kirsch W., Stollmann P., Stolz G.: Localization for random perturbations of periodic Schrödinger operators. Random Oper. Stoch. Eq. 6(3), 241–268 (1998)MATHMathSciNetGoogle Scholar
  23. 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)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Kitagaki Y.: Wegner estimates for some random operators with Anderson-type surface potentials. Math. Phys. Anal. Geom. 13(1), 47–67 (2010)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Kitagaki Y.: Generalized eigenvalue-counting estimates for some random acoustic operators. Kyoto J. Math. 51(2), 439–465 (2011)CrossRefMATHMathSciNetGoogle Scholar
  26. 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. 27.
    Klein A., Nguyen S.T.: The bootstrap multiscale analysis for the multi-particle Anderson model. J. Stat. Phys. 151(5), 938–973 (2013)ADSCrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Klopp F.: Localization for some continuous random Schrödinger operators. Commun. Math. Phys. 167(3), 553–569 (1995)ADSCrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Klopp F., Pankrashkin K.: Localization on quantum graphs with random vertex couplings. J. Stat. Phys. 131, 651–673 (2008)ADSCrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Klopp F., Pankrashkin K.: Localization on quantum graphs with random edge lengths. Lett. Math. Phys. 87, 99–114 (2009)ADSCrossRefMATHMathSciNetGoogle Scholar
  31. 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.Google Scholar
  32. 32.
    Krüger H.: Localization for random operators with non-monotone potentials with exponentially decaying correlations. Ann. Henri Poincaré 13(3), 543–598 (2012)ADSCrossRefMATHGoogle Scholar
  33. 33.
    Kuchment P.: Quantum graphs I. Some basic structures. Waves Random Media 14(1), S107–S128 (2004)ADSCrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Kunz H., Souillard B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 78(2), 201–246 (1980)ADSCrossRefMATHMathSciNetGoogle Scholar
  35. 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)CrossRefMATHMathSciNetGoogle Scholar
  36. 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)Google Scholar
  37. 37.
    Reed M., Simon B.: Methods of modern mathematical physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978)MATHGoogle Scholar
  38. 38.
    Rojas-Molina, C.: The Anderson model with missing sites. preprint arXiv: 1302.3640 (2013)Google Scholar
  39. 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)ADSCrossRefMATHGoogle Scholar
  40. 40.
    Sabri M., Étude de la localisation pour des systèmes désordonnés sur un graphe quantique. PhD Thesis, Univ. Paris Diderot (in preparation)Google Scholar
  41. 41.
    Simon B.: Lifschitz tails for the Anderson model. J. Stat. Phys. 38(1–2), 65–76 (1985)ADSCrossRefGoogle Scholar
  42. 42.
    Stollmann P.: Wegner estimates and localization for continuum Anderson models with some singular distributions. Arch. Math. (Basel) 75(4), 307–311 (2000)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Stollmann P.: From uncertainty principles to Wegner estimates. Math. Phys. Anal. Geom. 13(2), 145–157 (2010)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Veselić I.: Wegner estimate for discrete alloy-type models. Ann. Henri Poincaré 11(5), 991–1005 (2010)ADSCrossRefMATHGoogle Scholar
  45. 45.
    Veselić I.: Wegner estimates for sign-changing single site potentials. Math. Phys. Anal. Geom. 13(4), 299–313 (2010)CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Wegner F.: Bounds on the density of states in disordered systems. Z. Phys. B 44(1-2), 9–15 (1981)ADSCrossRefMathSciNetGoogle Scholar
  47. 47.
    Weidmann J.: Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68. Springer, New York (1980)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuUniversité Paris Diderot Paris 7ParisFrance
  2. 2.Department of Mathematics, Faculty of ScienceCairo UniversityCairoEgypt

Personalised recommendations