Global Poissonian behavior of the eigenvalues and localization centers of random operators in the localized phase

Frédéric Klopp[1]

  • [1] LAGA, U.M.R. 7539 C.N.R.S Institut Galilée Université Paris-Nord 99 Avenue J.-B. Clément F-93430 Villetaneuse France

Séminaire Laurent Schwartz — EDP et applications (2011-2012)

  • Volume: 2011-2012, page 1-12
  • ISSN: 2266-0607

Abstract

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In the present note, we review some recent results on the spectral statistics of random operators in the localized phase obtained in [12]. For a general class of random operators, we show that the family of the unfolded eigenvalues in the localization region considered jointly with the associated localization centers is asymptotically ergodic. This can be considered as a generalization of [10]. The benefit of the present approach is that one can vary the scaling of the unfolded eigenvalues covariantly with that of the localization centers. The convergence result then holds for all the scales that are asymptotically larger than the localization scale. We also provide a similar result that is localized in energy. Full proofs of the results presented here will be published elsewhere ([12]).

How to cite

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Klopp, Frédéric. "Global Poissonian behavior of the eigenvalues and localization centers of random operators in the localized phase." Séminaire Laurent Schwartz — EDP et applications 2011-2012 (2011-2012): 1-12. <http://eudml.org/doc/251174>.

@article{Klopp2011-2012,
abstract = {In the present note, we review some recent results on the spectral statistics of random operators in the localized phase obtained in [12]. For a general class of random operators, we show that the family of the unfolded eigenvalues in the localization region considered jointly with the associated localization centers is asymptotically ergodic. This can be considered as a generalization of [10]. The benefit of the present approach is that one can vary the scaling of the unfolded eigenvalues covariantly with that of the localization centers. The convergence result then holds for all the scales that are asymptotically larger than the localization scale. We also provide a similar result that is localized in energy. Full proofs of the results presented here will be published elsewhere ([12]).},
affiliation = {LAGA, U.M.R. 7539 C.N.R.S Institut Galilée Université Paris-Nord 99 Avenue J.-B. Clément F-93430 Villetaneuse France},
author = {Klopp, Frédéric},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {random operators; eigenvalues; localization centers},
language = {eng},
pages = {1-12},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Global Poissonian behavior of the eigenvalues and localization centers of random operators in the localized phase},
url = {http://eudml.org/doc/251174},
volume = {2011-2012},
year = {2011-2012},
}

TY - JOUR
AU - Klopp, Frédéric
TI - Global Poissonian behavior of the eigenvalues and localization centers of random operators in the localized phase
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2011-2012
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2011-2012
SP - 1
EP - 12
AB - In the present note, we review some recent results on the spectral statistics of random operators in the localized phase obtained in [12]. For a general class of random operators, we show that the family of the unfolded eigenvalues in the localization region considered jointly with the associated localization centers is asymptotically ergodic. This can be considered as a generalization of [10]. The benefit of the present approach is that one can vary the scaling of the unfolded eigenvalues covariantly with that of the localization centers. The convergence result then holds for all the scales that are asymptotically larger than the localization scale. We also provide a similar result that is localized in energy. Full proofs of the results presented here will be published elsewhere ([12]).
LA - eng
KW - random operators; eigenvalues; localization centers
UR - http://eudml.org/doc/251174
ER -

References

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  8. Werner Kirsch. An invitation to random Schrödinger operators. In Random Schrödinger operators, volume 25 of Panor. Synthèses, pages 1–119. Soc. Math. France, Paris, 2008. With an appendix by Frédéric Klopp. Zbl1162.82004MR2509110
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  10. Frédéric Klopp. Asymptotic ergodicity of the eigenvalues of random operators in the localized phase. To appear in Prob. Theor. Relat. Fields ArXiv: http://fr.arxiv.org/abs/1012.0831, 2010. Zbl1281.82014
  11. Frédéric Klopp. Inverse tunneling estimates and applications to the study of spectral statistics of random operators on the real line. ArXiv: http://fr.arxiv.org/abs/1101.0900, 2011. Zbl1293.35337MR2775121
  12. Frédéric Klopp. Universal joint asymptotic ergodicity of the eigenvalues and localization centers of random operators in the localized phase. In preparation, 2012. Zbl1337.60145
  13. Nariyuki Minami. Theory of point processes and some basic notions in energy level statistics. In Probability and mathematical physics, volume 42 of CRM Proc. Lecture Notes, pages 353–398. Amer. Math. Soc., Providence, RI, 2007. Zbl1137.81018MR2352280
  14. Leonid Pastur and Alexander Figotin. Spectra of random and almost-periodic operators, volume 297 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1992. Zbl0752.47002MR1223779
  15. Peter Stollmann. Caught by disorder, volume 20 of Progress in Mathematical Physics. Birkhäuser Boston Inc., Boston, MA, 2001. Bound states in random media. Zbl0983.82016MR1935594
  16. Ivan Veselić. Existence and regularity properties of the integrated density of states of random Schrödinger operators, volume 1917 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2008. Zbl1189.82004MR2378428

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