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Register a new userHolder regularity of the integrated density of states for quasi-periodic long-range operators on $ell^2(Z^d)$
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We prove the Holder continuity of the integrated density of states for a class of quasi-periodic long-range operators on $ell^2(Z^d)$ with large trigonometric polynomial potentials and Diophantine frequencies. Moreover, we give the Holder exponent in terms of the cardinality of the level sets of the potentials, which improves, in the perturbative regime, the result obtained by Goldstein and Schlag cite{gs2}. Our approach is a combination of Aubry duality, generalized Thouless formula and the regularity of the Lyapunov exponents of analytic quasi-periodic $GL(m,C)$ cocycles which is proved by quantitative almost reducibility method.
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For non-critical almost Mathieu operators with Diophantine frequency, we establish exponential asymptotics on the size of spectral gaps, and show that the spectrum is homogeneous. We also prove the homogeneity of the spectrum for Schodinger operators with (measure-theoretically) typical quasi-periodic analytic potentials and fixed strong Diophantine frequency. As applications, we show the discrete version of Deifts conjecture cite{Deift, Deift17} for subcritical analytic quasi-periodic initial data and solve a series of open problems of Damanik-Goldstein et al cite{BDGL, DGL1, dgsv, Go} and Kotani cite{Kot97}.
In this paper, we prove a power-law version dynamical localization for a random operator $mathrm{H}_{omega}$ on $mathbb{Z}^d$ with long-range hopping. In breif, for the linear Schrodinger equation $$mathrm{i}partial_{t}u=mathrm{H}_{omega}u, quad u in ell^2(mathbb{Z}^d), $$ the Sobolev norm of the solution with well localized initial state is bounded for any $tgeq 0$.
The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus $gge 2$. These flows are weakly mixing by the Avila-Forni theorem. In genus 2, the Holder property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained Holder estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Fornis idea with the symbolic approach of [10] and prove Holder regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera.
We show that a generic quasi-periodic Schrodinger operator in $L^2(mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrodinger operator with the resulting potential has empty absolutely continuous spectrum.
We consider the scattering of $n$ classical particles interacting via pair potentials, under the assumption that each pair potential is long-range, i.e. being of order ${cal O}(r^{-alpha})$ for some $alpha >0$. We define and focus on the free region, the set of states leading to well-defined and well-separated final states at infinity. As a first step, we prove the existence of an explicit, global surface of section for the free region. This surface of section is key to proving the smoothness of the map sending a point to its final state and to establishing a forward conjugacy between the $n$-body dynamics and free dynamics.
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