Skip to main content
SpringerLink
Log in

Conditional Wegner Estimate for the Standard Random Breather Potential

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We prove a conditional Wegner estimate for Schrödinger operators with random potentials of breather type. More precisely, we reduce the proof of the Wegner estimate to a scale free unique continuation principle. The relevance of such unique continuation principles has been emphasized in previous papers, in particular in recent years. We consider the standard breather model, meaning that the single site potential is the characteristic function of a ball or a cube. While our methods work for a substantially larger class of random breather potentials, we discuss in this particular paper only the standard model in order to make the arguments and ideas easily accessible.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Broderix, K., Hundertmark, D., Leschke, H.: Continuity properties of Schrödinger semigroups with magnetic fields. Rev. Math. Phys. 12(2), 181–225 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. 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  MATH  MathSciNet  Google Scholar 

  3. 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 

  4. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)

    Google Scholar 

  5. 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). arXiv:math-ph/0412078

  6. Hundertmark, D., Simon, B.: An optimal \({L}^p\)-bound on the Krein spectral shift function. J. Anal. Math. 87,199–208 (2002). http://www.ma.utexas.edu/mp_arc/c/00/00-370.ps.gz

  7. Hundertmark, D., Simon, B.: A diamagnetic inequality for semigroup differences. J. Reine Angew. Math. 571, 107–130 (2004)

    MATH  MathSciNet  Google Scholar 

  8. Kirsch, W., Veselić, I.: Lifshitz tails for a class of Schrödinger operators with random breather-type potential. Lett. Math. Phys. 94(1), 27–39 (2010)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. 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)

    Article  MATH  ADS  Google Scholar 

  10. Klopp, F., Loss, M., Nakamura, S., Stolz, G.: Localization for the random displacement model. Duke Math. J. 161(4), 587–621 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Leinfelder, H.: Gauge invariance of Schrödinger operators and related spectral properties. J. Oper. Theory 9, 163–179 (1983)

    MATH  MathSciNet  Google Scholar 

  12. Nakić, I., Täufer, M., Tautenhahn, M., Veselić, I.: Scale-free uncertainty principles and applications. Working paper, Technische Universität Chemnitz

  13. Nakić, I., Täufer, M., Tautenhahn, M., Veselić, I.: Scale-free uncertainty principles and Wegner estimates for random breather potentials. Comptes Rendus Mathematique (2014, to appear). arXiv:1410.5273

  14. Rojas-Molina, C., Veselić, I.: Scale-free unique continuation estimates and applications to random Schrödinger operators. Commun. Math. Phys. 320(1):245–274 (2013). arXiv:1110.4652

  15. Schumacher, C., Veselić, I.: Lifschitz tails for random breather potentials. Working paper, Technische Universität Chemnitz

  16. Simon, B.: Functional Integration and Quantum Physics. Academic Press Inc., New York (1979)

    MATH  Google Scholar 

  17. Veselić, I.: Lifshitz asymptotics for Hamiltonians monotone in the randomness. Oberwolfach Rep. 4(1):380–382 (2007). arXiv:0708.0487

  18. Veselić, I.: Existence and Regularity Properties of the Integrated Density of States of Random Schrödinger Operators. Lecture Notes in Mathematics. Springer, New York (2008)

    MATH  Google Scholar 

  19. Yafaev, D.: Mathematical Scattering Theory. Translations of Mathematical Monographs. American Mathematical Society, Providence (1992). Russian original: Izdatel’ stvo Sankt-Peterburgskogo Universiteta, St. Petersburg, 1994

    Google Scholar 

Download references

Acknowledgments

This work has been partially supported by the DFG under grant Eindeutige Fortsetzungsprinzipien und Gleichverteilungseigenschaften von Eigenfunktionen. It has also profited from interactions with Francisco Hoecker-Escuti, Ivica Nakić, Martin Tautenhahn and Christoph Schumacher. Part of these interactions have been supported by the binational German-Croatian DAAD project Scale-uniform controllability of partial differential equations.

Author information

Authors and Affiliations

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

    Matthias Täufer & Ivan Veselić

Authors
  1. Matthias Täufer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ivan Veselić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Veselić.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Täufer, M., Veselić, I. Conditional Wegner Estimate for the Standard Random Breather Potential. J Stat Phys 161, 902–914 (2015). https://doi.org/10.1007/s10955-015-1358-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-015-1358-y

Keywords

Search

Navigation