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Exercises for a Book on Random Potentials

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Book cover Random Media

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

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Abstract

We first review known results on the spectral properties of one dimensional random Schrodinger operators and of their perturbations by deterministric potentials. We then give some new results in the form of exercises and we then apply them to the case of multidimensional spherically symmetric random potentials.

Partially supported by NSF DMS 850–3695

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References

  1. F. Bentosela, R. Carmona, P. Ouclos, R. Simon, R. Souillard, R. Weder: Schrodinger Operators with an Electric Field and Random or Deterministic Potentials. Comm. Math. Phys. 88 (1983) 387–397.

    Google Scholar 

  2. J. Brossard: Perturbations Aleatoires de Potentiels Periodiques (preprint) Sept. 1982.

    Google Scholar 

  3. P. Carmona: Exponential Localization in One-Dimensional Disordered System, Duke Math. J. 49 (1982) 191–213.

    Google Scholar 

  4. R. Carmona: One dimensional Schrödinger Operators with Random or Deterministic Potentials: New Spectral Types, J. Functional Anal. 51 (1983) 229–258.

    Google Scholar 

  5. R. Carmona: Random Schrödinger Operators. Ecole d’Eté de Probabilités. Saint Flour (1984) to appear in Lect. Notes in Math. Springer Verlag.

    Google Scholar 

  6. E. A. Coddington, N. Levinson: Theory of Ordinary Differential Equations, McGraw Hill, New York (1955).

    Google Scholar 

  7. F. Deylon, H. Kunz, B. Souillard: One dimensional wave equations in random media, J. Phys. A16 (1983), 25–42.

    Google Scholar 

  8. I. Ya. Goldsheid, S. A. Molcanov, L. A. Pastur: A pure point spectrum of the stochastic one-dimensional Schrodinger operator. Funct. Analysis Appl. 11 (1977), 1–10.

    Google Scholar 

  9. L. Hörmander: Hypoelliptic differential equations of second order, Acta Math. 119 (1967) 147–171.

    Google Scholar 

  10. K. Ichihara, H. Kunita: A Classification of the Second Order Degenerate Elliptic Operators and its Probability Characterization. Z. Wahrscheinl. verw. Geb. 30 91974), 235–254.

    Google Scholar 

  11. S. Kotani: Lyapunov exponents and spectra for one-dimensional random Schrodinger operators. Proc. A.M.S. Conf. on “Random Matrices and their Applications”, June 1984.

    Google Scholar 

  12. S. A. Molcanov: The structure of eigenfunctions of one dimensional unordered structures, Math. U.S.S.R. Izvestija 12-1 (1978), 69–101.

    Google Scholar 

  13. M. Reed, B. Simon: Methods of Modern Mathematical Physics II, Fourier Analysis-Self-Adjointness, Academic Press (1975) New York.

    Google Scholar 

  14. D. Stroock, S. R. S. Varadhan: Multidimensional Diffusion Processes, Springer Verlag, (1979).

    Google Scholar 

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© 1987 Springer-Verlag New York, Inc.

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Carmona, R. (1987). Exercises for a Book on Random Potentials. 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_4

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  • DOI: https://doi.org/10.1007/978-1-4613-8725-1_4

  • 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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