TAK-875 clinical Simon discusses this in the remarks and his
Simon discusses this in the remarks and historical notes section following the general discussion of dynamically defined Verblunsky coefficients; see [12, pp. 706–707]. He points out that “understanding almost periodic Verblunsky coefficients is an intriguing open area” and in particular describes what one should expect in the small coupling regime, as well as the difficulties one faces when trying to establish a localization result for quasi-periodic Verblunsky coefficients.
A large deviation estimate In this paper we use the method developed by Jean Bourgain and Michael Goldstein in  to prove Anderson localization, which means that a large deviation estimate (LDT) for analytic quasi-periodic Szegő cocycles will play an important role. The first LDT for quasi-periodic Schrödinger operators was obtained in  and used there to prove Anderson localization. In this section, which is based on the more recent [14, Theorem 1], we will state an LDT for analytic quasi-periodic Szegő cocycles that is stronger than that in  and hence sufficient to prove Anderson localization. Fix an irrational and consider the continued fraction expansion with convergents for . Let Since is analytic on , it TAK-875 clinical has a bounded extension to a complex strip , . Define its norm by and set
Estimating transfer matrices and Green\'s functions In this section we give various bounds on the norms of n-step transfer matrices. First we establish a relation between n-step transfer matrices and the corresponding Lyapunov exponents. Then, by the method developed by Helge Krüger in , we can get that the entries of the Green\'s function decay exponentially off the diagonal, which is essential for proving Anderson localization.
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 , 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 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