All couplings localization for quasiperiodic operators with monotone potentials

Abstract

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.

Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles

“…They showed that the Lyapunov exponent is smooth or even analytic, respectively. Besides, for a 1-periodic function satisfying a Lipschitz monotonicity condition, Jitomirskaya and Kachkovskiy [25] showed that the Lyapunov exponent is almost Lipschitz continuous.…”

Dynamical Bounds for Quasiperiodic Schrödinger Operators with Rough Potentials

“…Another immediate corollary can be obtained for a class of discontinuous f monotone on the period as considered in [15]. Then, Anderson localization is established in [15] for Diophantine ω, while continuity (and positivity for large λ) of the Lyapunov exponent is established for all ω. Theorem 1.7 immediately implies in this case vanishing of β − for λ as above and all θ and all ω.…”

“…The latter is a general result that is of independent interest and of the type that has been crucial in various proofs of localization/regularity in many recent articles. Our extension has already been used in [15] for their spectral localization theorem and in [21] for their dimensional analysis of Sturmian potentials.…”

“…Remark 1.6 A large class of examples of operators(1.1) with discontinuous f and absolutely continuous N is presented in [15] Other potentials have been shown to satisfy the conditions of Theorem 1.1 in various regimes in [3,6,25,35]. In all those cases Theorem 1.1 improves on some of the known results since, even if the results potentially allowed for dynamical extensions, unlike Theorem 1.1 spectral localization cannot hold for all θ at least for continuous f that are even on the hull [19].…”

Power-Law Localization for Almost-periodic Operators: A Nash-Moser Iteration Approach

“…A natural generation of the potential x mod 1 is the so called Lipschitz monotone potential, which was introduced recently by [JK19]. In [JK19], the authors proved all couplings localization for the 1D Schrödinger operator with a Lipschitz monotone potential. • By δ ij we denote the Kronecker delta.…”