Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent

Bucaj, Valmir; Damanik, David; Fillman, Jake; Gerbuz, Vitaly; VandenBoom, Tom; Wang, Fengpeng; Zhang, Zhenghe

doi:10.1090/tran/7832

Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
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by Valmir Bucaj, David Damanik, Jake Fillman, Vitaly Gerbuz, Tom VandenBoom, Fengpeng Wang and Zhenghe Zhang PDF

Trans. Amer. Math. Soc. 372 (2019), 3619-3667

Abstract:

We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg’s theorem. That is, a Schrödinger operator in $\ell ^2(\mathbb {Z})$ whose potential is given by independent, identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions, and its unitary group exhibits exponential off-diagonal decay, uniformly in time. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogues of these models.

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