Abstract
In this note, we prove Minami’s estimate for a class of discrete alloy-type models with a sign-changing single-site potential of finite support. We apply Minami’s estimate to prove Poisson statistics for the energy level spacing. Our result is valid for random potentials which are in a certain sense sufficiently close to the standard Anderson potential (rank one perturbations coupled with i.i.d. random variables).
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Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)
Aizenman M.: Localization at weak disorder: some elementary bounds. Rev. Math. Phys. 6(5a), 1163–1182 (1994)
Aizenman M., Elgart A., Naboko S., Schenker J.H., Stolz G.: Moment analysis for localization in random Schrödinger operators. Invent. Math. 163(2), 343–413 (2006)
Aizenman M., Molchanov S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157(2), 245–278 (1993)
Anderson P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109(5), 1492–1505 (1958)
Bellissard J.V., Hislop P.D., Stolz G.: Correlations estimates in the lattice Anderson model. J. Stat. Phys. 129(4), 649–662 (2007)
Cao Z., Elgart A.: Weak localization for the alloy-type Anderson model on a cubic lattice. J. Stat. Phys. 148(6), 1006–1039 (2012)
Combes J.-M., Germinet F., Klein A.: Generalized eigenvalue-counting estimates for the Anderson model. J. Stat. Phys. 135(2), 201–216 (2009)
Combes J.-M., Germinet F., Klein A.: Poisson statistics for eigenvalues of continuum random Schrödinger operators. Anal. PDE 3(1), 49–80 (2010)
del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: What is localization? Phys. 551 Rev. Lett. 75(1), 117–119 (1995)
Disertori, M., Rivasseau, V.: Random matrices and the Anderson model. In: Random Schrödinger operators, Panor. Synthèses, vol. 25, pp. 161–213. Soc. Math. France, Paris (2008)
Efetov K.: Supersymmetry in Disorder and Chaos. Cambridge University Press, Cambridge (1997)
Elgart, A., Shamis, M., Sodin, S.: Localisation for non-monotone Schrödinger operators. arXiv:1201.2211v3[math-ph] (2012)
Elgart A., Tautenhahn M., Veselić I.: Localization via fractional moments for models on \({\mathbb{Z}}\) with single-site potentials of finite support. J. Phys. A Math. Theor. 43(47), 474021 (2010)
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 12(8), 1571–1599 (2011)
Fröhlich J., Martinelli F., Scoppola E., Spencer T.: Constructive proof of localization in the Anderson tight binding model. Commun. Math. Phys. 101(1), 21–46 (1985)
Fröhlich J., Spencer T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88(2), 151–184 (1983)
Germinet F., Klein A.: A characterization of the Anderson metal-insulator transport transition. Duke Math. J. 124(2), 309–350 (2004)
Germinet, F., Klopp, F.: Spectral statistics for random Schrödinger operators in the localized regime. J. Eur. Math. Soc. arXiv:1011.1832v3[math.SP] (2012)
Germinet, F., Klopp, F.: Enhanced Wegner and Minami estimates and eigenvalue statistics of random Anderson models at spectral edges. Ann. Henri Poincaré (2012). doi:10.1007/s00023-012-0217-5
Goldsheid I.Y., Molchanov S., Pastur L.: A pure point spectrum of the stochastic one-dimensional Schrödinger operator. Funct. Anal. Appl. 11(1), 1–10 (1977)
Graf G.M.: Anderson localization and the space-time characteristic of continuum states. J. Stat. Phys. 75(1-2), 337–346 (1994)
Graf G.M., Vaghi A.: A remark on the estimate of a determinant by Minami. Lett. Math. Phys. 79(1), 17–22 (2007)
Hislop P.D., Klopp F.: The integrated density of states for some random operators with nonsign definite potentials. J. Funct. Anal. 195(1), 12–47 (2002)
Hislop P.D., Müller P.: A lower bound for the density of states of the lattice Anderson model. Proc. Am. Math. Soc. 136(8), 2887–2893 (2008)
Jeske, F.: Über lokale Positivität der Zustandsdichte zufälliger Schrödinger-Operatoren. Ph.D. thesis, Ruhr Universität Bochum, Germany (1992)
Leonhardt, K., Peyerimhoff, N., Tautenhahn, M., Veselić, I.: Wegner estimate and localisation for alloy-type models with sign-changing exponentially decaying single-site potentials. TU Chemnitz (2013, preprint)
Killip R., Nakano F.: Eigenfunction statistics in the localized Anderson model. Ann. Henri Poincaré 8(1), 27–36 (2007)
Kirsch W., Stollmann P., Stolz G.: Anderson localization for random Schrödinger operators with long range interactions. Commun. Math. Phys. 195(3), 495–507 (1998)
Klein, A.: Multiscale analysis and localization of random operators. In: Random Schrödinger operators, Panoramas et synthèses, vol. 25, pp. 121–159. Société Mathématique de France (2008)
Klein A., Molchanov S.: Simplicity of eigenvalues in the Anderson model. J. Stat. Phys. 122(1), 95–99 (2006)
Klopp F.: Localization for some continuous random Schrödinger operators. Commun. Math. Phys. 167(3), 553–569 (1995)
Klopp F.: Weak disorder localization and Lifshitz tails: continuous Hamiltonians. Ann. Henri Poincaré 3(4), 711–737 (2002)
Klopp, F.: Inverse tunneling estimates and applications to the study of spectral statistics of random operators on the real line. J. Reine Angew. Math. (2012). doi:10.1515/crelle-2012-0026
Klopp F., Nakamura S.: Lifshitz tails for generalized alloy-type random Schrödinger operators. Anal. PDE 3(4), 409–426 (2010)
Krüger H.: Localization for random operators with non-monotone potentials with exponentially decaying correlations. Ann. Henri Poincaré 13(3), 543–598 (2012)
Minami N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177(3), 709–725 (1996)
Molchanov S.: The local structure of the spectrum of the one-dimensional Schrödinger operator. Commun. Math. Phys. 78(3), 429–446 (1981)
Molchanov S.: The structure of eigenfunctions of one-dimensional unordered structures. Math. USSR Izvestija 12(69), 69–101 (1978)
Nakano F.: The repulsion between localization centers in the Anderson model. J. Stat. Phys. 123(4), 803–810 (2006)
Nakano F.: The distribution of localization centers in some discrete random systems. Rev. Math. Phys. 19(9), 941–965 (2007)
Peyerimhoff, N., Tautenhahn, M., and Veselić, I.: Wegner estimate for alloy-type models with sign-changing and exponentially decaying single-site potentials. Technische Universität Chemnitz, Preprintreihe der Fakultät für Mathematik, preprint 2011-9, ISSN 1614-8835 (2011)
Režnikova A.J.: The central limit theorem for the spectrum of the random one-dimensional Schrödinger operator. J. Stat. Phys. 25(2), 291–308 (1981)
Tautenhahn, M., and Veselić, I.: A note on regularity for discrete alloy-type models. Technische Universität Chemnitz, Preprintreihe der Fakultät für Mathematik, preprint 2010-6, ISSN 1614-8835 (2010)
Tautenhahn, M., Veselić, I.: Spectral properties of discrete alloy-type models. In: Exner, P. (ed.) XVIth International Congress on Mathematical Physics, pp. 551–555 (2010)
Veselić, I.: Indefinite Probleme bei der Anderson-Lokalisierung. Ph.D. thesis, Ruhr-Universität Bochum, January (2001)
Veselić I.: Wegner estimate and the density of states of some indefinite alloy type Schrödinger operators. Lett. Math. Phys. 59(3), 199–214 (2002)
Veselić I.: Wegner estimate for discrete alloy-type models. Ann. Henri Poincaré 11(5), 991–1005 (2010)
Veselić I.: Wegner estimates for sign-changing single site potentials. Math. Phys. Anal. Geom. 13(4), 299–313 (2010)
Veselić I.: Lipschitz-continuity of the integrated density of states for Gaussian random potentials. Lett. Math. Phys. 97(1), 25–27 (2011)
von Dreifus H., Klein A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124(2), 285–299 (1989)
Wegner F.: Bounds on the DOS in disordered systems. Z. Phys. B 44, 9–15 (1981)
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Communicated by Anton Bovier.
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Tautenhahn, M., Veselić, I. Minami’s Estimate: Beyond Rank One Perturbation and Monotonicity. Ann. Henri Poincaré 15, 737–754 (2014). https://doi.org/10.1007/s00023-013-0263-7
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DOI: https://doi.org/10.1007/s00023-013-0263-7