Duke Mathematical Journal

Localization for the random displacement model

Frédéric Klopp, Michael Loss, Shu Nakamura, and Günter Stolz

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We prove spectral and dynamical localization for the multidimensional random displacement model near the bottom of its spectrum by showing that the approach through multiscale analysis is applicable. In particular, we show that a previously known Lifshitz tail bound can be extended to our setting and prove a new Wegner estimate. A key tool is given by a quantitative form of a property of a related single-site Neumann problem which can be described as “bubbles tend to the corners.”

Article information

Duke Math. J., Volume 161, Number 4 (2012), 587-621.

First available in Project Euclid: 1 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 47B80: Random operators [See also 47H40, 60H25]


Klopp, Frédéric; Loss, Michael; Nakamura, Shu; Stolz, Günter. Localization for the random displacement model. Duke Math. J. 161 (2012), no. 4, 587--621. doi:10.1215/00127094-1548353. https://projecteuclid.org/euclid.dmj/1330610808

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  • [1] J. Baker, M. Loss, and G. Stolz, Minimizing the ground state energy of an electron in a randomly deformed lattice, Comm. Math. Phys. 283 (2008), 397–415.
  • [2] J. Baker, M. Loss, and G. Stolz, Low energy properties of the random displacement models, J. Funct. Anal. 256 (2009), 2725–2740.
  • [3] M. Sh. Birman and D. R. Yafaev, The spectral shift function: The work of M. G. Krein and its further development, St. Petersburg Math. J. 4 (1993), 833–870.
  • [4] D. Buschmann and G. Stolz, Two-parameter spectral averaging and localization properties for non-monotonic random Schrödinger operators, Trans. Amer. Math. Soc. 353 (2000), 635–653.
  • [5] J. M. Combes, P. D. Hislop, and S. Nakamura, The Lp-theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators, Comm. Math. Phys. 218 (2001), 113–130.
  • [6] D. Damanik, D. Lenz, and G. Stolz, Lower transport bounds for one-dimensional continuum Schrödinger operators, Math. Ann. 336 (2006), 361–389.
  • [7] D. Damanik, R. Sims, and G. Stolz, Localization for one-dimensional continuum, Bernoulli-Anderson models, Duke Math. J. 114 (2002), 59–100.
  • [8] D. Damanik and S. Tcheremchantsev, Power-law bounds on transfer matrices and quantum dynamics in one dimension, Comm. Math. Phys. 236 (2003), 513–534.
  • [9] R. Fukushima, Brownian survival and Lifshitz tail in perturbed lattice disorder, J. Funct. Anal. 256 (2009), 2867–2893.
  • [10] R. Fukushima and N. Ueki, Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice, Ann. Henri Poincaré 11 (2010), 1053–1083.
  • [11] F. Germinet, P. D. Hislop, and A. Klein, Localization at low energies for attractive Poisson random Schrödinger operators, CRM Proc. Lecture Notes 42 (2007), 153–165.
  • [12] F. Germinet, P. D. Hislop and A. Klein, Localization for Schrödinger operators with Poisson random potential, J. Eur. Math. Soc. 9 (2007), 577–607.
  • [13] F. Germinet and A. Klein, Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222 (2001), 415–448.
  • [14] F. Ghribi and F. Klopp, Localization for the random displacement model at weak disorder, Ann. Henri Poincaré 11 (2010), 127–149.
  • [15] P. D. Hislop and F. Klopp, The integrated density of states for some random operators with nonsign definite potentials, J. Funct. Anal. 195 (2002), 12–47.
  • [16] S. Jitomirskaya, H. Schulz-Baldes, and G. Stolz, Delocalization in random polymer models, Comm. Math. Phys. 233 (2003), 27–48.
  • [17] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, Berlin, 1976.
  • [18] A. Klein, Multiscale analysis and localization of random operators. Random Schröinger operators, Panor. Synthèses 25 (2008), 121–159.
  • [19] F. Klopp, Localization for semiclassical continuous random Schrödinger operators, II: The random displacement model, Helv. Phys. Acta 66 (1993), 810–841.
  • [20] F. Klopp and S. Nakamura, Spectral extrema and Lifshitz tails for non monotonous alloy type models, Comm. Math. Phys. 287 (2009), 1133–1143.
  • [21] F. Klopp and S. Nakamura, Lifshitz tails for generalized alloy type random Schrödinger operators, Analysis and PDE 3 (2010), 409–426.
  • [22] F. Klopp, S. Nakamura, F. Nakano, and Y. Nomura, Anderson localization for 2D discrete Schrödinger operators with random magnetic fields, Ann. H. Poincaré 4 (2003), 795–811.
  • [23] B. Simon, Spectral averaging and the Krein spectral shift, Proc. Amer. Math. Soc. 126 (1998), 1409–1413.
  • [24] R. Sims, Localization in one-dimensional models of disordered media, Ph.D. dissertation, University of Alabama at Birmingham, 2001.
  • [25] P. Stollmann, Caught by Disorder: Bound States in Random Media, Prog. Math. Phys. 20, Birkhäuser, Boston, 2001.
  • [26] D. R. Yafaev, Mathematical Scattering Theory: General Theory, Transl. Math. Monogr. 105, Amer. Math. Soc., Providence, 1992.