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Browsing Mathematics Department by Subject "CMV matrices"
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Item Anderson localization for quasi-periodic CMV matrices and quantum walks(Elsevier, 2019) Wang, Fengpeng; Damanik, DavidWe consider CMV matrices, both standard and extended, with analytic quasi-periodic Verblunsky coefficients and prove Anderson localization in the regime of positive Lyapunov exponents. This establishes the CMV analog of a result Bourgain and Goldstein proved for discrete one-dimensional Schrödinger operators. We also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi-periodic coins.Item Generic spectral results for CMV matrices with dynamically defined Verblunsky coefficients(Elsevier, 2020) Fang, Licheng; Damanik, David; Guo, ShuzhengWe consider CMV matrices with dynamically defined Verblunsky coefficients. These coefficients are obtained by continuous sampling along the orbits of an ergodic transformation. We investigate whether certain spectral phenomena are generic in the sense that for a fixed base transformation, the set of continuous sampling functions for which the spectral phenomenon occurs is residual. Among the phenomena we discuss are the absence of absolutely continuous spectrum and the vanishing of the Lebesgue measure of the spectrum.