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Random Media pp 245–266Cite as

Regularity of the Density of States for Stochastic Jacobi Matrices: A Review

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Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 7))

Abstract

In this paper, we will discuss stochastic Jacobi matrices which are operators on l 2 (Zυ). Indicate elements of this Hilbert space by u(n) with nZv.

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References

  1. Avron, J., Simon, B., Almost periodic Schrodinger operators, I. Limit periodic potentials, Comm. Math. Phys. 82 (1982), 101–120

    Article  MathSciNet  Google Scholar 

  2. Avron, J., Simon, B., Almost periodic Schrödinger operators, II. The integrated density of states, Duke Math. J. 50 (1983), 369–391

    Article  MathSciNet  MATH  Google Scholar 

  3. Bellissard, J., Lima, R., Testard, D., Almost periodic Schrödinger operators, ZIF preprint

    Google Scholar 

  4. Bellissard, J., Lima, R., Testard, D., A metal-insulator transition for the almost Mathieu model, Comm. Math. Phys. 88 (1983), 207–234

    Article  MathSciNet  MATH  Google Scholar 

  5. Bellissard, J. and Simon, B., Cantor spectrum for the almost Mathieu equation, J. Func. Anal. 48 (1982), 408–419

    Article  MathSciNet  MATH  Google Scholar 

  6. Benderskii, M., Pastur, L., On the spectrum of the one-dimensional Schrödinger equation with a random potential, Math. USSR Sb. 11 (1970), 245

    Article  Google Scholar 

  7. Campanino, M., Klein, A., A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model, Comm. Math. Phys., to appear

    Google Scholar 

  8. Carmona, R., Lectures on random Schrödinger operators, to appear in Proc. 14th St. Flour Probability Summer School (1985), Springer

    Google Scholar 

  9. Chulaevsky, V., On perturbations of a Schrödinger operator with periodic potential, Russian Math. Surveys 36 (1981), 143

    Article  Google Scholar 

  10. Constantinescu, F., Fröhlich, J., Spencer, T., Analyticity of the density of states and replica method for random Schrödinger operators on a lattice, J. Stat. Phys. 34 (1984), 571–596

    Article  MATH  Google Scholar 

  11. Craig, W., Pure point spectrum for discrete almost periodic Schrödinger operators, Comm. Math. Phys. 88 (1983), 113–131

    Article  MathSciNet  MATH  Google Scholar 

  12. Craig, W. and Simon, B., Subharmonicity of the Lyaponov index, Duke Math. J. 50 (1983), 551–560

    Article  MathSciNet  MATH  Google Scholar 

  13. Craig, W., Simon, B., Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices, Comm. Math. Phys. 90 (1983), 207–218

    Article  MathSciNet  MATH  Google Scholar 

  14. Cycon, H., Froese, R., Kirsch, W., Simon, B., Topics in the theory of Schrodinger operators, to appear

    Google Scholar 

  15. Delyon, F., Souillard, B., The rotation number for finite difference operators and its properties, Comm. Math. Phys. 89 (1983), 415

    Article  MathSciNet  MATH  Google Scholar 

  16. Delyon, F., Souillard, B., Remark on the continuity of the density of states of ergodic finite difference operators, Comm. Math. Phys. 94 (1984), 289

    Article  MathSciNet  MATH  Google Scholar 

  17. Endrullis, M., Englisch, H., Special energies and special frequencies, preprint

    Google Scholar 

  18. Grempel, D., Fishman, S., Prange, R., Localization in an incomemensurate potential: An exactly solvable model, Phys. Rev. Lett. 49 (1982), 833

    Article  MathSciNet  Google Scholar 

  19. Halperin, B., Properties of a particle in a one-dimensional random potential, Adv. Chem. Phys. 31 (1967), 123–177

    Article  Google Scholar 

  20. Johnson, R., Moser, J., The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), 403

    Article  MathSciNet  MATH  Google Scholar 

  21. Kirsch, W., Martinelli, F., Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians, Comm. Math. Phys. 89 (1983), 27–40

    Article  MathSciNet  MATH  Google Scholar 

  22. Kirsch, W., Simon, B., Lifshitz tails for periodic plus random potentials, J. Stat. Phys., to appear

    Google Scholar 

  23. Kotani, S., Ljaponov indices determine absolutely continuous spectra of stationary random one-dimensional Schrodinger operators, Stochastic analysis (1984), K. Ito (ed.), North Holland, pp. 225–248

    Google Scholar 

  24. Kunz, H., Souillard, B., Sur le spectre des operateurs aux differences finies aleatoires, Comm. Math. Phys. 78 (1980), 201–246

    Article  MathSciNet  MATH  Google Scholar 

  25. LePage, E., Empirical distribution of the eigenvalues of a Jacobi matrix, Probability measures on groups, VII, Springer Lecture Notes Series 1064, 1983, pp. 309–367

    Google Scholar 

  26. LePage, E., in prep.

    Google Scholar 

  27. Lifschitz, I., Energy spectrum structure and quantum states of disordered condensed systems, Sov. Phys. Usp. 7 (1965), 549

    Article  Google Scholar 

  28. Lloyd, P., Exactly solvable model of electronic states in a three-dimensional disordered Hamiltonian: Non-existence of localized states, J. Phys. C2 (1969), 1717–1725

    Google Scholar 

  29. Luck, J., Nieuwenhuizen, T., Singular behavior of the density of states and the Lyaponov coefficient in binary random harmonic chains, J. Stat. Phys., to appear

    Google Scholar 

  30. March, P. and Snitzman, A., Some connections between excursion theory and the discrete random Schrodinger equation, with applications to analyticity and smoothness properties of the density of states in one dimension, Courant preprint

    Google Scholar 

  31. Mezincescu, G., Bounds on the integrated density of electronic states for disordered Hamiltonians, Phys. Rev., to appear

    Google Scholar 

  32. Mezincescu, G., Internal Lifschitz singularities of disordered finite-difference Schrodinger operators. Comm. Math. Phys., to appear

    Google Scholar 

  33. Moser, J., An example of a Schrödinger equation with an almost periodic potential and nowhere dense spectrum, Comm. Math. Helv. 56 (1981), 198

    Article  MATH  Google Scholar 

  34. Pastur, L., Spectral properties of disordered systems in one-body approximation, Comm. Math. Phys. 75 (1980), 179

    Article  MathSciNet  MATH  Google Scholar 

  35. Poschel, J., Examples of discrete Schrodinger operators with pure point spectrum, Comm. Math. Phys. 88 (1983), 447–463

    Article  MathSciNet  Google Scholar 

  36. Simon, B., Kotani theory for one-dimensional stochastic Jacobi matrices, Comm. Math. Phys. 89 (1983), 227

    Article  MathSciNet  MATH  Google Scholar 

  37. Simon, B., The equality of the density of states in a wide class of tight binding Lorentzian models, Phys. Rev. B27 (1983), 3859–3860

    Google Scholar 

  38. Simon, B., Almost periodic Schrödinger operators, IV. The Maryland model, Ann. Phys. 159 (1985), 157–183

    Article  MATH  Google Scholar 

  39. Simon, B., Lifschitz tails for the Anderson model, J. Stat. Phys. 38 (1985), 65–76

    Article  Google Scholar 

  40. Simon, B., Internal Lifschitz singularities in the Anderson model, in prep.

    Google Scholar 

  41. Simon, B., Taylor, M., Harmonic analysis on SL(2,R) and smoothness of the density of states in the one dimensional Anderson model, Comm. Math. Phys., to appear

    Google Scholar 

  42. Simon, B., Wolff, T., Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math., to appear

    Google Scholar 

  43. Spencer, T., The Schrödinger equation with a random potential–a mathematical review, to appear in Proc. 1984 Les Houches Summer School

    Google Scholar 

  44. Wannier, G., Claro, F., Magnetic subband structure of electrons in hexagonal lattices, Phys. Rev. B19 (1979), 6068

    Google Scholar 

  45. Wegner, F., Bounds on the density of states in disordered systems, Z. Phys. B44 (1981), 9–15.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

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

    Barry Simon

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  1. Barry Simon
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Editor information

Editors and Affiliations

  1. Institute for Mathematics and Its Applications, Minneapolis, Minnesota, 55455, USA

    George Papanicolaou

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© 1987 Springer-Verlag New York, Inc.

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Simon, B. (1987). Regularity of the Density of States for Stochastic Jacobi Matrices: A Review. In: Papanicolaou, G. (eds) Random Media. The IMA Volumes in Mathematics and Its Applications, vol 7. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8725-1_17

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  • DOI: https://doi.org/10.1007/978-1-4613-8725-1_17

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8727-5

  • Online ISBN: 978-1-4613-8725-1

  • eBook Packages: Springer Book Archive

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