Titlebook: Spectral Theory of Random Schr?dinger Operators; René Carmona,Jean Lacroix Book 1990 Birkh?user Boston 1990 Finite.H?lder condition.Identi

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ETCH

Ergodic Families of Self-Adjoint Operators,of these possibly unbounded operators. We believe that all the technical difficulties relative to these measurability problems have been carefully swept under the rug in most of the research litterature. This is one of the reasons why we decided to study these problems thoroughly.

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Localization in Any Dimension,ial interest. The links between the exponential growth of the solutions of the eigenvalue equation and the exponential decay of the Green’s function have already been pointed out in the one dimensional case.

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Localization in One Dimension, conjectured by Mott & Twose in [249] that this property should hold in the one dimensional case at any disorder. This chapter is devoted to the proof of this last conjecture which we will extend to quasi-one dimensional systems.

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Book 1990ng, the phenomenon can be described as follows: the self-adjoint operators which are used as Hamiltonians for these systems have a ten- dency to have pure point spectrum, especially in low dimension or for large disorder. A lot of effort has been devoted to the mathematical study of the random self-

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Spectral Theory of Self-Adjoint Operators,ded in the sequel. Because of lack of space, we refrain from explaining the motivations behind the numerous definitions we introduce. We merely illustrate them with examples of Schr?dinger operators and we postpone a more detailed study to Chapter II. Rather than a serious introduction to the spectr

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,Schr?dinger Operators,te the abstract theory of Chapter I by the concrete examples of the generalized Laplacian operators in Section II.1 and their perturbations by multiplication operators in Section II.2. The latter are of special importance since they are the Schr?dinger operators we want to study. We consider the pro

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Products of Random Matrices,t important result in this direction is the extension to matrix valued random variables of the strong law of large numbers. Unfortunately the identification of the limit (called the Lyapunov exponent) is more complicated than in the classical case of real valued random variables. In particular this