Abstract
We describe a program initiated by Moser [1], and Johnson-Moser [2], and then developed by Kotani [3] (Kotani's work was extended to the discrete case by Simon [4]) and Deift-Simon [5]. We discuss relations between the density of states, the Lyaponov exponent, and the classical m-function of Weyl. In particular, we obtain Kotani's results that the essential support of the absolutely continuous spectrum is precisely the set where the Lyaponov exponent vanishes, and that in the random non-deterministic case the Lyaponov exponent is a.e. positive. We also describe the Deift-Simon results that, in the discrete case, the Lebesgue measure of the set where the Lyaponov exponent vanishes is at most 4, and the construction of continuum eigenfunctions for the absolutely continuous spectrum.
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