Abstract
We consider the random operator: −d/m ω(dx)d +/dx+q ω(x), wherem ω(dx) andq ω(x) are a stationary ergodic random measure and a random function respectively. To this general case, we extend Kotani's theorem which asserts that the absolutely continuous spectrum is completely determined by the Ljapounov indices. Our framework includes the case of stochastic Jacobi matrices treated by Simon.
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Carmona, R.: Random Schrödinger operators. École d'Été de Probabilités de Saint-Flour 1984 (to appear)
Cornfeld, I. P., Fomin, S. V., Sinai, Ya. G.: Ergodic theory. Berlin, Heidelberg, New York: Springer 1982
Craig, W., Simon, B.: Subharmonicity of Ljapounov index. Duke Math. J.50, No. 2, 551–560 (1983)
Deift, P., Simon, B.: Almost periodic Schrödinger operators III. The absolutely continuous spectrum in one dimension. Commun. Math. Phys.90, 389–411 (1983)
Dym, H., McKean, H. P.: Gaussian processes, function theory and the inverse spectral problems. New York, San Francisco, London: Academic Press 1976
Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys.84, 403–438 (1982)
Kac, I. S., Krein, M. G.: R-functions-analytic functions maping the upper halfplane into itself. Am. Math. Soc. Trans. (2)103, 1–18
Kingman, J. F. C.: Subadditive processes. École d'Été de Probabilités de Saint-Flour 1975. Lecture Notes in Mathematics, vol.539, pp. 167–223 Berlin, Heidelberg, New York: Springer 1976
Kotani, S.: Ljapounov indices determine absolutely continuous spectra of random one-dimensional Schrödinger operators. Proc. Taniguchi Symp. SA Katata (1982), 225–247
Kotani, S., Watanabe, S.: Krein's spectral theory of strings and generalized diffusion processes. Lecture Notes in Mathematics, vol.923, pp. 235–259 Berlin, Heidelberg, New York: Springer 1982
McKean, H. P.: Elementary solutions for certain parabolic partial differential equations. Tran. Am. Math. Soc.82, 519–548 (1956)
Pastur, L. A.: Spectral properties of disordered systems in the one-body approximation. Commun. Math. Phys.75, 179–196 (1980)
Simon, B.: Kotani theory for one dimensional stochastic Jacobi matrices. Commun. Math. Phys.89, 227–234 (1983)
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Minami, N. An extension of Kotani's theorem to random generalized Sturm-Liouville operators. Commun.Math. Phys. 103, 387–402 (1986). https://doi.org/10.1007/BF01211754
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DOI: https://doi.org/10.1007/BF01211754