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We give a criterion based on reflection symmetries in the spirit of Jitomirskaya–Simon to show absence of point spectrum for (split-step) quantum walks and Cantero–Moral–Velázquez (CMV) matrices. To accomplish this, we use some ideas from a recent paper by the authors and collaborators to implement suitable reflection symmetries for such operators. We give several applications. For instance, we deduce arithmetic delocalization in the phase for the unitary almost-Mathieu operator and singular continuous spectrum for generic CMV matrices generated by the Thue–Morse subshift.
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From the general inverse theory of periodic Jacobi matrices, it is known that a periodic Jacobi matrix of minimal period p≥2 may have at most p−2 closed spectral gaps. We discuss the maximal number of closed gaps for one-dimensional periodic discrete Schrödinger operators of period p. We prove nontrivial upper and lower bounds on this quantity for large p and compute it exactly for p≤6. Among our results, we show that a discrete Schrödinger operator of period four or five may have at most a single closed gap, and we characterize exactly which potentials may exhibit a closed gap. For period six, we show that at most two gaps may close. In all cases in which the maximal number of closed gaps is computed, it is seen to be strictly smaller than p−2, the bound guaranteed by the inverse theory. We also discuss similar results for purely off-diagonal Jacobi matrices.more » « less
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Abstract We investigate the symmetries of the so-called generalized extended Cantero–Moral–Velázquez (CMV) matrices. It is well-documented that problems involving reflection symmetries of standard extended CMV matrices can be subtle. We show how to deal with this in an elegant fashion by passing to the class of generalized extended CMV matrices via explicit diagonal unitaries in the spirit of Cantero–Grünbaum–Moral–Velázquez. As an application of these ideas, we construct an explicit family of almost-periodic CMV matrices, which we call the mosaic unitary almost-Mathieu operator, and prove the occurrence of exact mobility edges. That is, we show the existence of energies that separate spectral regions with absolutely continuous and pure point spectrum and exactly calculate them.more » « less
