Duke Mathematical Journal

Generic Singular Spectrum For Ergodic Schrödinger Operators

Artur Avila and David Damanik

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Abstract

We consider Schrödinger operators with ergodic potential $V_\omega(n)=f(T^{n}(\omega))$, $n \in \bb{Z}$, $\omega \in \Omega$, where $T:\Omega \to \Omega$ is a nonperiodic homeomorphism. We show that for generic $f \in C(\Omega)$, the spectrum has no absolutely continuous component. The proof is based on approximation by discontinuous potentials which can be treated via Kotani theory

Article information

Source
Duke Math. J. Volume 130, Number 2 (2005), 393-400.

Dates
First available: 15 November 2005

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1132064631

Digital Object Identifier
doi:10.1215/S0012-7094-05-13035-6

Mathematical Reviews number (MathSciNet)
MR2181094

Subjects
Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations

Citation

Avila, Artur; Damanik, David. Generic Singular Spectrum For Ergodic Schrödinger Operators. Duke Mathematical Journal 130 (2005), no. 2, 393--400. doi:10.1215/S0012-7094-05-13035-6. http://projecteuclid.org/euclid.dmj/1132064631.


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References

  • A. Avila and J. Bochi, A formula with some applications to the theory of Lyapunov exponents, Israel J. Math. 131 (2002), 125--137.
  • A. Avila and R. Krikorian, Quasiperiodic $\SL(2,\R)$ cocycles, in preparation.
  • J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems 22 (2002), 1667--1696.
  • J. Bourgain and S. Jitomirskaya, Absolutely continuous spectrum for $1$D quasiperiodic operators, Invent. Math. 148 (2002), 453--463.
  • H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts Monogr. Phys., Springer, Berlin, 1987.
  • D. Damanik and R. Killip, Ergodic potentials with a discontinuous sampling function are nondeterministic, Math. Res. Lett. 12 (2005), 187, --192.
  • P. Deift and B. Simon, Almost periodic Schrödinger operators, III: The absolutely continuous spectrum in one dimension, Comm. Math. Phys. 90 (1983), 389--411.
  • T. A. Driscoll and L. N. Trefethen, Schwarz-Christoffel Mapping, Cambridge Monogr. Appl. Comput. Math. 8, Cambridge Univ. Press, Cambridge, 2002.
  • M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5--233.
  • K. Ishii, Localization of eigenstates and transport phenomena in one-dimensional disordered systems, Suppl. Prog. Theoret. Phys. 53 (1973), 77--138.
  • S. Kotani, ``Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators'' in Stochastic Analysis (Katata/Kyoto, 1982), North-Holland Math. Library 32, North-Holland, Amsterdam, 1984, 225--247.
  • —, Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys. 1 (1989), 129--133.
  • H. Kunz and B. Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys. 78 (1980/81), 201--246.
  • D. Lenz and P. Stollmann, Generic sets in spaces of measures and generic singular continuous spectrum for Delone Hamiltonians, to appear in Duke Math. J., preprint.
  • L. A. Pastur, Spectral properties of disordered systems in one-body approximation, Comm. Math. Phys. 75 (1980), 179--196.
  • B. Simon, Kotani theory for one-dimensional stochastic Jacobi matrices, Comm. Math. Phys. 89 (1983), 227--234.
  • —, Operators with singular continuous spectrum, I: General operators, Ann. of Math. (2) 141 (1995), 131--145.