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Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schr\"odinger operators

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arxiv 1510.07086 v2 pith:7OEC5L6W submitted 2015-10-23 math.SP

Svetlana Jitomirskaya , Shiwen Zhang This is my paper · ORCID
classification math.SP
keywords almostarithmeticspectralapplicationsbetacriterialyapunovodinger
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abstract

We introduce a notion of $\beta$-almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded $\beta$-almost periodic potentials. Applications include a sharp arithmetic criterion of full spectral dimensionality for analytic quasiperiodic Schr\"odinger operators in the positive Lyapunov exponent regime and arithmetic criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator.

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  1. Anderson localization for multi-frequency quasi-periodic operators on $\mathbb{Z}^d$

    math.SP 2019-08 conditional novelty 7.0 of 10

    For analytic multi-frequency quasi-periodic operators on Z^d, Anderson localization holds at strong coupling for arbitrary number of frequencies and dimension when the phase space dimension is at least d.

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