Abstract. We give a simple argument that if a quasiperiodic multi-frequency Schrödinger cocycle is reducible to a constant rotation for almost all energies with respect to the density of states measure, then the spectrum of the dual operator is purely point for Lebesgue almost all values of the ergodic parameter θ. The result holds in the L 2 setting provided, in addition, that the conjugation preserves the fibered rotation number. Corollaries include localization for (long-range) 1D analytic potentials with dual ac spectrum and Diophantine frequency as well as a new result on multidimensional localization.
For a wide class of 2D periodic elliptic operators, we show that the global extrema of all spectral band functions are isolated.
Abstract. We consider isotropic XY spin chains whose magnetic potentials are quasiperiodic and the effective one-particle Hamiltonians have absolutely continuous spectra. For a wide class of such XY spin chains, we obtain lower bounds on their Lieb-Robinson velocities in terms of group velocities of their effective Hamiltonians:where E is considered as a function of the integrated density of states.
We establish Anderson localization for quasiperiodic operator families of the formfor all λ > 0 and all Diophantine α, provided that v is a 1-periodic function satisfying a Lipschitz monotonicity condition on [0, 1). The localization is uniform on any energy interval on which Lyapunov exponent is bounded from below. 1 1 We employ the word non-perturbative in the widely used by now sense of "obtained as a corollary of positive Lyapunov exponents without further largeness/smallness assumptions" [9,28].2 Maryland model exhibits uniform localization for energies restricted to a finite interval, but not overall.3 Even though the technical analysis only holds for a sparse sequence of scales, this is sufficient for a conclusion on the IDS. This is what allows to obtain the result without any Diophantine conditions.
with the operator H(θ 1 , θ 2 ) satisfying Anderson localization in the region of energies E (2.4) |E − U j | λ e (log λ) µ , j = 1, . . . , N int , for all ω ∈ Ω(U, λ, θ 1 , θ 2 ) and for all possible translations of U.Remark 2.4.(1) The part of the spectrum removed by (2.4) is contained in N int intervals of size o(|λ|).(2) The condition on separability of v is irrelevant in Theorem 2.2. The potential v(θ 1 ) + v(θ 2 ) can be replaced by an analytic function w(•, •) ∈ C ω (T 2 ) of two variables that is not constant on any straight line segment, with the same proof. (3) An analogue of Theorem 2.3 can also be obtained for non-separable case, assuming |E − (λw i + U j )| λε for every w i such that w(θ 1 , θ 2 ) ≡ w i on some straight line segment, and every value U j of U, also with the same proof. (4) As discussed above, the inclusion of a background potential U of low complexity could have been done already in [9], as well as in the other papers that establish perturbative results by semi-algebraic techniques and do not involve Lyapunov exponents/cocycles (for example, in [10]). ( 5) Suppose that v satisfies Type II symmetry, U = 0, θ 1 = θ 2 = 1/4. Then one can easily check that ψ(n 1 , n 2 ) = (−1) n 1 δ n 1 n 2 solves the eigenvalue equation H(1/4, 1/4)ψ = 0. While this does not contradict purely point spectrum, all known proofs of Anderson localization show that any solution of the eigenvalue equation decays exponentially, which does not allow the existence of states like ψ. This example suggests that some stronger versions of localization can break down at zero energy, but only in symmetric cases (because otherwise Theorem 2.2 holds). Possible scenarios of delocalization at zero energy in different models are described in [20,15]. (6) The condition in Theorem 2.3 of analyticity of v in the strip of size 20 is technical and can possibly be removed with some extra work. (7) The operator family H(θ 1 , θ 2 ) is not ergodic because U is not assumed to have any translational invariance. However, one can prove localization simultaneously for all translations of U. (8) Our results are perturbative, in the sense that one always has to remove a positive measure set of frequencies. However, our requirements on the frequency are more explicit. The bound on λ 0 can be expressed, in principle, through the Diophantine constant C dio of ω (see (5.1)). The parameter ε freq in Theorems 2.2 and 2.3 is, essentially, the measure of frequencies for which (5.1) does not hold with this C dio . Afterwards, as usually happens in localization proofs, one has to remove an extra set of frequencies of measure zero, for which we do not have any arithmetic description, and which depends on θ 1 , θ 2 , λ, and other parameters. (9) Theorems 2.2 and 2.3 are formulated for the case of a single phase (θ 1 , θ 2 ).However, one can extend them for a full measure set of phases, see Remark 9.1. We do not believe this argument is new, however, in the case of perturbative results, it has not been explicitly stated in the literature.
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