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  • br Elimination of double resonances and


    Elimination of double resonances and semi-algebraic sets In this section we prove a statement that is usually referred to as the elimination of double resonances. From the proof of [3], one may see that the uniform positivity of and a uniform LDT for the Lyapunov exponent, together with the elimination of double resonances, imply Anderson localization.
    In fact, if we replace (4.11) by the condition and restrict the index set to rather than to , we can also get the estimate (4.13). Denote by the frequency set obtained in Lemma 4.4 and define Then we have Since is decreasing, we can take a countable subsequence of κ and hence obtain . Assume and let , satisfy the equation where Assume and . Since , there exists such that for all . Since , we can take large enough to satisfy Since , according to condition (4.10), then if there is such that Lemma 4.4 implies that for all , (4.12) must fail, that is, Then, by Lemma 3.5, there exist β and γ such that for each , Now consider the interval , where (this makes sure that ). Invoking the paving property (Appendix 7.2), we can deduce from (5.4) that Restricting the equation (5.1) to , for , we have (for details, see Appendix 7.1), and hence This is the required exponential decay property (also valid for the negative side). It remains to show that for some , the following inequality holds, Recalling , [10, Lemma 3.9] implies Thus it ITSA1 will suffice to show that there exists such that Let where . Assume for some , Then we have which implies for , Obviously, we also have . Similarly, assuming the same method yields . Since implies that (5.9) is satisfied, it remains to show that (5.10) and (5.11) hold. Letting , we verify them by averaging over . Thus recalling Lemma 3.3 (with n replaced by ), we see that Hence, there is such that implying by the upper bound (Lemma 3.2) Hence (5.10) holds, and in the same way we may obtain the estimate (5.11). Therefore exhibits Anderson localization for a.e. , assuming .
    Quantum walks A quantum walk is described by a unitary operator on the Hilbert space , which models a state space in which a wave packet comes equipped with a spin at each integer site. Here, the elementary tensors of the form and with comprise an orthonormal basis of (where denotes the canonical basis of ). A time-homogeneous quantum walk scenario is given as soon as unitary coins are specified. As one passes from time t to time , the update rule of the quantum walk applies the coins coordinate-wise and shifts spin-up states to the right and spin-down states to the left, viz If we extend this by linearity and continuity to general elements of , this defines a unitary operator U on . Equivalently, denote a typical element by where one must have We may then describe the action of U in coordinates via and the matrix representation of a quantum walk is given by Quantum walks can be connected to extended CMV matrices as follows. If all Verblunsky coefficients with even index vanish, then the extended CMV matrix becomes which resembles the matrix representation U.