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Kotani theory for one dimensional stochastic Jacobi matrices

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Abstract

We consider families of operators,H ω, on ℓ2 given by (H ω u)(n)=u(n+1)+u(n−1)+V ω(n)u(n), whereV ω is a stationary bounded ergodic sequence. We prove analogs of Kotani's results, including that for a.e. ω,σac(H ω) is the essential closure of the set ofE where γ(E) the Lyaponov index, vanishes and the result that ifV ω is non-deterministic, then σac is empty.

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References

  1. Avron, J., Simon, B.: Almost periodic Schrodinger operators, II. The integrated density of states. Duke Math. J. (to appear)

  2. Carmona, R.: One dimensional Schrondinger operators with random or deterministic potentials: New spectral types. J. Funct. Anal.51, (1983)

  3. Casher, A., Lebowitz, J.: Heat flow in disordered harmonic chains. J. Math. Phys.12, 8 (1971)

    Google Scholar 

  4. Craig, W., Simon, B.: Subharmonicity of the Lyaponov index. Duke Math. J. (submitted)

  5. Delyon, F., Souillard, B.: The rotation number for finite difference operators and its properties. Commun. Math. Phys. (to appear)

  6. Donoghue, W.: Distributions and Fourier transforms. New York: Academic Press 1969

    Google Scholar 

  7. Ishii, K.: Localization of eigenstates and transport phenomena in the one dimensional disordered system. Supp. Theor. Phys.53, 77–138 (1973)

    Google Scholar 

  8. Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys.84, 403–438 (1982)

    Google Scholar 

  9. Katznelson, Y.: An introduction to harmonic analysis. New York: Dover 1976

    Google Scholar 

  10. Kotani, S.: Lyaponov indices determine absolutely continuous spectra of stationary random one-dimensional Schrondinger operators. Proc. Kyoto Stoch. Conf., 1982

  11. Pastur, L.: Spectral properties of disordered systems in the one body approximation. Commun. Math. Phys.75, 179–196 (1980)

    Google Scholar 

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Authors and Affiliations

  1. Departments of Mathematics and Physics, California Institute of Technology, 91125, Pasadena, CA, USA

    Barry Simon

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  1. Barry Simon
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Communicated by T. Spencer

Research partially supported by USNSF under Grant MCS-81-20833

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Simon, B. Kotani theory for one dimensional stochastic Jacobi matrices. Commun.Math. Phys. 89, 227–234 (1983). https://doi.org/10.1007/BF01211829

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  • DOI: https://doi.org/10.1007/BF01211829

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