Duke Mathematical Journal

An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators

Jean-Michel Combes, Peter D. Hislop, and Frédéric Klopp

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove that the integrated density of states (IDS) of random Schrödinger operators with Anderson-type potentials on L2(Rd) for d1 is locally Hölder continuous at all energies with the same Hölder exponent 0<α1 as the conditional probability measure for the single-site random variable. As a special case, we prove that if the probability distribution is absolutely continuous with respect to Lebesgue measure with a bounded density, then the IDS is Lipschitz continuous at all energies. The single-site potential uL0(Rd) must be nonnegative and compactly supported. The unperturbed Hamiltonian must be periodic and satisfy a unique continuation principle (UCP). We also prove analogous continuity results for the IDS of random Anderson-type perturbations of the Landau Hamiltonian in two dimensions. All of these results follow from a new Wegner estimate for local random Hamiltonians with rather general probability measures

Article information

Source
Duke Math. J. Volume 140, Number 3 (2007), 469-498.

Dates
First available in Project Euclid: 8 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1194547696

Digital Object Identifier
doi:10.1215/S0012-7094-07-14032-8

Mathematical Reviews number (MathSciNet)
MR2362242

Zentralblatt MATH identifier
1134.81022

Subjects
Primary: 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 47B80: Random operators [See also 47H40, 60H25] 60H25: Random operators and equations [See also 47B80] 35P05: General topics in linear spectral theory

Citation

Combes, Jean-Michel; Hislop, Peter D.; Klopp, Frédéric. An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J. 140 (2007), no. 3, 469--498. doi:10.1215/S0012-7094-07-14032-8. https://projecteuclid.org/euclid.dmj/1194547696


Export citation

References

  • M. Aizenman, A. Elgart, S. Naboko, J. H. Schenker, and G. Stolz, Moment analysis for localization in random Schrödinger operators, Invent. Math. 163 (2006), 343--413.
  • J.-M. Barbaroux, J. M. Combes, and P. D. Hislop, Localization near band edges for random Schrödinger operators, Helv. Phys. Acta 70 (1997), 16--43.
  • R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Probab. Appl., Birkhaüser, Boston, 1990.
  • J.-M. Combes and P. D. Hislop, Localization for some continuous, random Hamiltonians in d-dimensions, J. Funct. Anal. 124 (1994), 149--180.
  • —, Landau Hamiltonians with random potentials: Localization and the density of states, Comm. Math. Phys. 177 (1996), 603--629.
  • J.-M. Combes, P. D. Hislop, and F. Klopp, Hölder continuity of the integrated density of states for some random operators at all energies, Int. Math. Res. Not. 2003, no. 4, 179--209.
  • —, ``Some new estimates on the spectral shift function associated with random Schrödinger operators'' in Probability and Mathematical Physics (Montréal, 2005), CRM Proc. Lecture Notes 42, Amer. Math. Soc., Providence, 2007, 85--96.
  • J.-M. Combes, P. D. Hislop, F. Klopp, and G. Raikov, Global continuity of the integrated density of states for random Landau Hamiltonians, Comm. Partial Differential Equations 29 (2004), 1187--1213.
  • J.-M. Combes, P. D. Hislop, and E. Mourre, Spectral averaging, perturbation of singular spectra, and localization, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4883--4894.
  • J.-M. Combes, P. D. Hislop, and S. Nakamura, The $L^p$-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.
  • J.-M. Combes, P. D. Hislop, and A. Tip, Band edge localization and the density of states for acoustic and electromagnetic waves in random media, Ann. Inst. H. Poincaré Phys. Théor. 70 (1999), 381--428.
  • S.-I. Doi, A. Iwatsuka, and T. Mine, The uniqueness of the integrated density of states for the hr operators with magnetic fields, Math. Z. 237 (2001), 335--371.
  • A. Figotin and A. Klein, Localization of classical waves, I: Acoustic waves, Comm. Math. Phys. 180 (1996), 439--482.
  • —, Localization of classical waves, II: Electromagnetic waves, Comm. Math. Phys. 184 (1997), 411--441.
  • W. Fischer, T. Hupfer, H. Leschke, and P. MüLler, Existence of the density of states for multi-dimensional continuum hr operators with Gaussian random potentials, Comm. Math. Phys. 190 (1997), 133--141.
  • F. Germinet and A. Klein, Operator kernel estimates for functions of generalized Schrödinger operators, Proc. Amer. Math. Soc. 131 (2003), 911--920.
  • F. Germinet, A. Klein, and J. H. Schenker, Dynamical delocalization in random Landau Hamiltonians, to appear in Ann. of Math. (2), preprint,\arxivmath-ph/0412070v1
  • D. Hundertmark, R. Killip, S. Nakamura, P. Stollmann, and I. Veselić, Bounds on the spectral shift function and the density of states, Comm. Math. Phys. 262 (2006), 489--503.
  • D. Hundertmark and B. Simon, A diamagnetic inequality for semigroup differences, J. Reine Angew. Math. 571 (2004), 107--130.
  • T. Hupfer, H. Leschke, P. MüLler, and S. Warzel, The absolute continuity of the integrated density of states for magnetic hr operators with certain unbounded random potentials, Comm. Math. Phys. 221 (2001), 229--254.
  • —, Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials, Rev. Math. Phys. 13 (2001), 1547--1581.
  • W. Kirsch, ``Random hr operators: A course'' in hr Operators (Sønderborg, Denmark, 1988), Lecture Notes in Phys. 345, Springer, Berlin, 1989, 264--370.
  • S. Kotani and B. Simon, Localization in general one-dimensional random systems, II: Continuum Schrödinger operators, Comm. Math. Phys. 112 (1987), 103--119.
  • S. N. Naboko, The structure of singularities of operator functions with a positive imaginary part (in Russian), Funktsional. Anal. i Prilozhen. 25, no. 4 (1991), 1--13.; English translation in Funct. Anal. Appl. 25 (1991), 243--253.
  • S. Nakamura, A remark on the Dirichlet-Neumann decoupling and the integrated density of states, J. Funct. Anal. 179 (2001), 136--152.
  • L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Grundlehren Math. Wiss. 297, Springer, Berlin, 1992.
  • B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Note Ser. 35, Cambridge Univ. Press, Cambridge, 1979.
  • P. Stollmann, Wegner estimates and localization for continuum Anderson models with some singular distributions, Arch. Math. (Basel) 75 (2000), 307--311.
  • —, Caught by Disorder: Bound States in Random Media, Prog. Math. Phys. 20, Birkhaüser, Boston, 2001.
  • B. Sz.-Nagy [SzöKefalvi-Nagy] and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970.
  • I. Veselić, Wegner estimate and the density of states of some indefinite alloy-type Schrödinger operators, Lett. Math. Phys. 59 (2002), 199--214.
  • W.-M. Wang, Microlocalization, percolation, and Anderson localization for the magnetic Schrödinger operator with a random potential, J. Funct. Anal. 146 (1997), 1--26.
  • —, Supersymmetry and density of states of the magnetic Schrödinger operator with a random potential revisited, Comm. Partial Differential Equations 25 (2000), 601--679.
  • F. Wegner, Bounds on the density of states in disordered systems, Z. Phys. B 44 (1981), 9--15.
  • T. H. Wolff, ``Recent work on sharp estimates in second order elliptic unique continuation problems'' in Fourier Analysis and Partial Differential Equations (Miraflores de la Sierra, Spain, 1992), Stud. Adv. Math., CRC, Boca Raton, Fla., 1995, 99--128.