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Kotani theory for one dimensional stochastic Jacobi matrices. (English) Zbl 0534.60057
S. Kotani [Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators. Proc. Kyoto Stoch. Conf. 1982] recently proved three remarkable theorem on one- dimensional Schrödinger operators with stochastic potentials. The paper under review proves the analogs of these theorems for stochastic Jacobi matrices (in physical terms: the tight binding approximation). The first two theorems say that the Lyapunov index \(\gamma\) (E) determines the absolutely continuous spectrum \(\sigma_{ac}\) of the operator. The third theorem states that \(\gamma>0\) and thus \(\sigma_{ac}=\emptyset\) if the potential \(V_{\omega}(n)\) is non-deterministic. The proof follows the strategy of Kotani’s paper, while a series of non trivial changements in the details is necessary.
Reviewer: W.Kirsch

60H25 Random operators and equations (aspects of stochastic analysis)
81Q99 General mathematical topics and methods in quantum theory
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