Center for diffusion of mathematic journals

 
 
 
 

Séminaire Laurent Schwartz — EDP et applications

Table of contents for this volume | Previous article | Next article
Frédéric Klopp
Global Poissonian behavior of the eigenvalues and localization centers of random operators in the localized phase
Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exp. No. 9, 12 p., doi: 10.5802/slsedp.7
Article PDF

Résumé - 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]).

Bibliography

[1] Michael Aizenman, Jeffrey H. Schenker, Roland M. Friedrich, and Dirk Hundertmark. Finite-volume fractional-moment criteria for Anderson localization. Comm. Math. Phys., 224(1):219–253, 2001. Dedicated to Joel L. Lebowitz.  MR 1868998 |  Zbl 1038.82038
[2] Jean-Michel Combes, François Germinet, and Abel Klein. Generalized eigenvalue-counting estimates for the Anderson model. J. Stat. Phys., 135(2):201–216, 2009.  MR 2505733 |  Zbl 1168.82016
[3] Jean-Michel Combes, François Germinet, and Abel Klein. Poisson statistics for eigenvalues of continuum random Schrödinger operators. Anal. PDE, 3(1):49–80, 2010.  MR 2663411 |  Zbl 1227.82034
[4] François Germinet and Abel Klein. A characterization of the Anderson metal-insulator transport transition. Duke Math. J., 124(2):309–350, 2004.  MR 2078370 |  Zbl 1062.82020
[5] Francois Germinet and Abel Klein. New characterizations of the region of complete localization for random Schrödinger operators. J. Stat. Phys., 122(1):73–94, 2006.  MR 2203782 |  Zbl 1127.82031
[6] François Germinet and Frédéric Klopp. Improved Wegner and Minami type estimates and applications to the spectral statistics of random anderson models in the localized regime. in progress.
[7] François Germinet and Frédéric Klopp. Spectral statistics for random Schrödinger operators in the localized regime. ArXiv http://arxiv.org/abs/1011.1832, 2010.
[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.  MR 2509110 |  Zbl 1162.82004
[9] Werner Kirsch and Bernd Metzger. The integrated density of states for random Schrödinger operators. In Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, volume 76 of Proc. Sympos. Pure Math., pages 649–696. Amer. Math. Soc., Providence, RI, 2007.  MR 2307751 |  Zbl 1208.82028
[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.
[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.  MR 2775121
[12] Frédéric Klopp. Universal joint asymptotic ergodicity of the eigenvalues and localization centers of random operators in the localized phase. In preparation, 2012.
[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.  MR 2352280 |  Zbl 1137.81018
[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.  MR 1223779 |  Zbl 0752.47002
[15] Peter Stollmann. Caught by disorder, volume 20 of Progress in Mathematical Physics. Birkhäuser Boston Inc., Boston, MA, 2001. Bound states in random media.  MR 1935594 |  Zbl 0983.82016
[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.  MR 2378428 |  Zbl 1189.82004
Copyright Cellule MathDoc 2019 | Credit | Site Map