Related papers: Reducibility without KAM

We consider one-frequency analytic SL(2,R) cocycles. Our main result establishes the Almost Reducibility Conjecture in the case of exponentially Liouville frequencies. Together with our earlier work, this implies that all cocycles close to…

In this paper, we prove the generic version of Cantor spectrum for quasi-periodic Schr\"{o}dinger operators with finitely smooth and small potentials, and we also show pure point spectrum for a class of multi-frequency $C^k$ long-range…

This paper establishes an extreme $C^k$ reducibility theorem of quasi-periodic $SL(2, \mathbb{R})$ cocycles in the local perturbative region, revealing both the essence of Eliasson [Commun.Math.Phys.1992] and Hou-You [Invent.Math.2012] in…

Quasi-periodic cocycles with a diophantine frequency and with values in SL(2,R) are shown to be almost reducible as long as they are close enough to a constant, in the topology of k times differentiable functions, with k great enough.…

Given a quasiperiodic cocycle in sl(2, R) sufficiently close to a constant, we prove that it is almost-reducible in ultradifferentiable class under an adapted arithmetic condition on the frequency vector. We also give a corollary on the…

We establish a quantitative version of strong almost reducibility result for $\mathrm{sl}(2,\mathbb{R})$ quasi-periodic cocycle close to a constant in Gevrey class. We prove that, for the quasi-periodic Schr\"odinger operators with small…

We show that a one-frequency analytic SL(2,R) cocycle with Diophantine rotation vector is analytically linearizable if and only if the Lyapunov exponent is zero through a complex neighborhood of the circle. More generally, we show (without…

We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-R{\"u}ssmann condition in the analytic…

In this paper, we show that one-dimensional discrete multi-frequency quasiperiodic Schr\"odinger operators with smooth potentials demonstrate ballistic motion on the set of energies on which the corresponding Schr\"odinger cocycles are…

We give a simple argument that if a quasiperiodic multi-frequency Schr\"odinger 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…

We show that for almost every frequency alpha \in \R \setminus \Q, for every C^omega potential v:\R/\Z \to R, and for almost every energy E the corresponding quasiperiodic Schrodinger cocycle is either reducible or nonuniformly hyperbolic.…

We develop a new KAM scheme that applies to SL(2,R) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg-Sinai's Theorem to arbitrary frequencies: under a…

In the present paper, we establish a reduction theorem for linear Schr\"odinger equation with finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition by means of KAM technique. Moreover, it is proved that the…

For quasiperiodic Schr\"odinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schr\"odinger cocycle is either rotations reducible or has positive…

We prove a reducibility result for a linear wave equation with a time quasi-periodic driving on the one dimensional torus. The driving is assumed to be fast oscillating, but not necessarily of small size. Provided that the external…

We eliminate by KAM methods the time dependence in a class of linear differential equations in $\ell^2$ subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator $…

In this paper we consider a class of fully nonlinear forced and reversible Schroedinger equations and prove existence and stability of quasi-periodic solutions. We use a Nash-Moser algorithm together with a reducibility theorem on the…

The arithmetics of the frequency and of the rotation number play a fundamental role in the study of reducibility of analytic quasi-periodic cocycles which are sufficiently close to a constant. In this paper we show how to generalize…

We consider continuous $\mathrm{SL}(2,\mathbb{R})$ valued cocycles over general dynamical systems and discuss a variety of uniformity notions. In particular, we provide a description of uniform one-parameter families of continuous…

Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schr\"odinger equation (NLS) on the two dimensional torus $\mathbb T^2:= (\mathbb R/2\pi \mathbb Z)^2$, we consider a quasi-periodically…