About
43
Publications
4,351
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
3,400
Citations
Introduction
Skills and Expertise
Publications
Publications (43)
An obituary for Alexander Gordon which will appear in the Journal of Spectral Theory
The extended Harper’s model, proposed by D.J. Thouless in 1983, generalizes the famous almost Mathieu operator, allowing for a wider range of lattice geometries (parametrized by three coupling parameters) by permitting 2D electrons to hop to both nearest and next nearest neighboring (NNN) lattice sites, while still exhibiting its characteristic sym...
The extended Harper's model, proposed by D.J. Thouless in 1983, generalizes the famous almost Mathieu operator, allowing for a wider range of lattice geometries (parametrized by three coupling parameters) by permitting 2D electrons to hop to both nearest and next nearest neighboring (NNN) lattice sites, while still exhibiting its characteristic sym...
Quasi-periodic Schr\"odinger-type operators naturally arise in solid state
physics, describing the influence of an external magnetic field on the
electrons of a crystal. In the late 1970s, numerical studies for the most
prominent model, the almost Mathieu operator (AMO), produced the first example
of a fractal in physics known as "Hofstadter's butt...
We show that on a dense open set of analytic one-frequency complex valued
cocycles in arbitrary dimension Oseledets filtration is either dominated or
trivial. The underlying mechanism is different from that of the Bochi-Viana
Theorem for continuous cocycles, which links non-domination with discontinuity
of the Lyapunov exponent. Indeed, in our sett...
We show how to extend (and with what limitations) Avila’s global theory of analytic SL(2,C) cocycles to families of cocycles with singularities. This allows us to develop a strategy to determine the Lyapunov exponent for the extended Harper’s model, for all values of parameters and all irrational frequencies. In particular, this includes the self-d...
We study discrete quasiperiodic Schrödinger operators on \({\ell^2(\mathbb{Z})}\) with potentials defined by γ-Hölder functions. We prove a general statement that for γ > 1/2 and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An impo...
We prove that the exponential moments of the position operator stay bounded
for the supercritical almost Mathieu operator with Diophantine frequency.
Consider a quasi-periodic Schr\"odinger operator $H_{\alpha,\theta}$ with
analytic potential and irrational frequency $\alpha$. Given any rational
approximating $\alpha$, let $S_+$ and $S_-$ denote the union, respectively, the
intersection of the spectra taken over $\theta$. We show that up to sets of
zero Lebesgue measure, the absolutely continuou...
We prove that the Lyapunov exponent of quasi-periodic cocycles with singularities behaves continuously over the analytic category.
We thereby generalize earlier results, where singularities were either excluded completely or constrained by additional hypotheses.
Applications include parameter dependent families of analytic Jacobi operators, such as...
We establish sharp results on the modulus of continuity of the distribution of the spectral measure for one-frequency Schrödinger
operators with Diophantine frequencies in the region of absolutely continuous spectrum. More precisely, we establish 1/2-Hölder
continuity near almost reducible energies (an essential support of absolutely continuous spe...
We show how to extend (and with what limitations) Avila's global theory of
analytic SL(2,C) cocycles to families of cocycles with singularities. This
allows us to develop a strategy to determine the Lyapunov exponent for extended
Harper's model, for all values of parameters and all irrational frequencies. In
particular, this includes the self-dual...
It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasiperiodic cocycles. In this paper we show that it is continuous in the analytic category. Our corollaries include continuity of the Lyapunov exponent associated with general quasiperiodic Jacobi matrices or orthogonal polynomials on the unit ci...
We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schr\"odinger cocycles associated to a non-perturbatively small analytic potential and Diophantine frequency. In particular, we establish the full version of Eliasson's reducibility theory in this regime (our approach actually leads t...
In this paper, we show how the methods from [B-G] may be adapted to establish Anderson localization for quasi-periodic lattice
Schrödinger operators corresponding to the band model ℤ × {1, ..., b}. Recall that ‘Anderson localization’ means pure point spectrum with exponentially decaying eigenfunctions. We also discuss
the issue of dynamical localiz...
We discuss the recent proof of Cantor spectrum for the almost Mathieu operator for all conjectured values of the parameters.
A method is presented for proving upper bounds on the moments of the position operator when the dynamics of quantum wavepackets is governed by a random (possibly correlated) Jacobi matrix. As an application, one obtains sharp upper bounds on the diffusion exponents for random polymer models, coinciding with the lower bounds obtained in a prior work...
We prove the conjecture (known as the ``Ten Martini Problem'' after Kac and
Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all
non-zero values of the coupling and all irrational frequencies.
We show strong dynamical localization for a family of one-dimensional quasiperiodic Jacobi operators of magnetic origin, throughout the regime of positive Lyapunov exponents.
A random polymer model is a one-dimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent van...
Study of fine spectral properties of quasiperiodic and similar discrete Schr\"odinger operators involves dealing with problems caused by small denominators, and until recently was only possible using perturbative methods, requiring certain small parameters and complicated KAM-type schemes. We review the recently developed nonperturbative methods fo...
A polymer model is a one-dimensional Schrodinger operator composed of two nite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. Although the random models are known to have pure-poi...
We study regularity properties of the Lyapunov exponent L of quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove that L is jointly continuous, in frequency and energy, at every irrational frequency.
We study discrete Schrodinger operators (H ; )(n) = (n 1) + (n + 1) + f(n+) (n) on l (Z), where f(x) is a real analytic periodic function of period 1. We prove a general theorem relating the measure of the spectrum of H ; to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of H ; ar...
In this note we prove Strong Dynamical Localization for the almost Mathieu operator Htheta,lambda,omega=-Delta+ lambdacos(2pi(theta + xomega)) for all lambda>2 and Diophantine frequencies omega. This improves the previous known result [22, 13] which established Dynamical Localization for a.e. theta and for lambda>=15.
this paper we study quasiperiodic operators H !;;` , acting on ` (Z) and given by ( ~ H !;;` Psi)(n) = Psi(n + 1) + Psi(n Gamma 1) + f(!n + `)Psi(n); (1.1) where f(`) is an analytic 1Gammaperiodic function, in the regime of small coupling (large ): Write f as f(x) = a k e 2ikx with a Gammak = a k and ja k j Ce Gammak ; ? 0: Without loss we assume a...
We prove that for quasiperiodic operators with potential V(n)=f(+n), f analytic, the spectral measures are zero-dimensional for large, any irrational . It extends a result of Jitomirskaya and Last to the case of any analytic f.
We study Hausdorff-dimensional spectral properties of certain “whole-line” quasiperiodic discrete Schrödinger operators by
using the extension of the Gilbert–Pearson subordinacy theory that we previously developed in [19].
We study Hausdorff-dimensional spectral properties of certain “whole-line” quasiperiodic discrete Schrodinger operators by using the extension of the Gilbert–Pearson subordinacy theory that we previously developed in [19].
We show that the almost Mathieu operator, (H !;;` Psi)(n) = Psi(n + 1) + Psi(n Gamma 1) + cos(ß!n + `)Psi(n), has semi-uniform (and thus dynamical) localization for ? 15 and a.e. !; `. We also obtain a new estimate on gap continuity (in !) for this operator with ? 29 (or ! 4=29), and use it to prove that the measure of its spectrum is equal to j4 G...
. In a variety of contexts, we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular spectrum are a dense G ffi . In the spectral analysis of various operators of mathematical physics, a key step, often the hardest, is to prove that the operator has no conti...
. We study the almost Mathieu operator (h ;ff;` u)(n) = u(n+ 1) +u(n Gamma 1) + cos(ßffn + `)u(n) on ` 2 (Z), and prove that the dual of point spectrum is absolutely continuous spectrum. We use this to show that for = 2 it has purely singular continuous spectrum for a.e. pairs (ff; `). The ff's for which we prove this are explicit. x1. Introduction...
We prove that one-dimensional Schrödinger operators with even almost periodic potential have no point spectrum for a denseG
δ in the hull. This implies purely singular continuous spectrum for the almost Mathieu equation for coupling larger than 2 and a denseG
δ in θ even if the frequency is an irrational with good Diophantine properties.
We prove that for any diophantine rotation angle ω and a.e. phase θ the almost Mathieu operator (H(θ)Ψ)n
=Ψ
n−1
+Ψ
n+1
+λcos(2π(θ+nω))Ψ
n
has pure point spectrum with exponentially decaying eigenfunctions for λ≧15. We also prove the existence of some pure point spectrum for any λ≧5.4.
In this paper we study localization for ergodic families of discrete Schrödinger operators. We prove that instability of pure point spectrum implies absence of uniform localization.
Although concrete operators with singular continuous spectrum have proliferated recently [7, 11, 13, 17, 34, 35, 37, 39], we still don't really understand much about singular continuous spectrum. In part, this is because it is normally defined by what it isn't─neither pure point nor absolutely continuous. An important point of view, going back in p...
We examine various issues relevant to localization in the Anderson
model. We show there is more to localization than exponentially
localized states by presenting an example with such states but where
<x\(t\)2>/t2-δ is unbounded for any
δ>0. We show that the recently discovered instability of
localization under rank one perturbations is only a weak...
We prove that for any [^(s)]\hat \sigma
of the spectrum. Corresponding eigenfunctions decay exponentially. The singular continuous component, if it exists, is concentrated on a set of zero measure which is nowhere dense in
[^(s)]\hat \sigma
.
In a variety of contexts, we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular spectrum are a dense $G_\delta$.
Quantum Ising models in a transverse field are related to continuous-time percolation processes whose oriented percolation versions are contact processes. We study such models in the presence of quasiperiodic disorder and prove localization in the ground state, no percolation, and extinction, respectively, for sufficiently large disorder.
The talks will be devoted to the recent solutions of two problems of the 21st century.* Both problems are related to the almost Mathieu operator. The recent advances utilize at least partially the dynamical approach. We will discuss the necessary preliminaries and outline the ideas of the proofs. The talks are based on a paper by A. Avila, R. Kriko...