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.
Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schr\"odinger operators
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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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Anderson localization for multi-frequency quasi-periodic operators on $\mathbb{Z}^d$
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.