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Svetlana Jitomirskaya

Publications and source records attributed to Svetlana Jitomirskaya.

At least 37 records · Page 2Linked to original sources
arXiv2019

Inhomogeneous Diophantine approximation in the coprime setting

Given $n\in N$ and $x,γ\in R$, let \begin{equation*} ||γ-nx||^\prime=\min\{|γ-nx+m|:m\in Z, \gcd (n,m)=1\}, \end{equation*} %where $(n,m)$ is the largest common divisor of $n$ and $m$. Two conjectures in the coprime inhomogeneous Diophantine approximation state that for any irrational number $α$ and almost every $γ\in R$, \begin{equation*} \liminf_{n\to \infty}n||γ-nα||^{\prime}=0 \end{equation*} and that there exists $C>0$, such that for all $α\in R\backslash Q$ and $γ\in [0,1)$ , \begin{equation*} \liminf_{n\to \infty}n||γ-nα||^{\prime} < C. \end{equation*} We prove the first conjecture and disprove the second one.

math.NT↗
arXiv2018

Large deviations of the Lyapunov exponent and localization for the 1D Anderson model

The proof of Anderson localization for the 1D Anderson model with arbitrary (e.g. Bernoulli) disorder, originally given by Carmona-Klein-Martinelli in 1987, is based in part on the multi-scale analysis. Later, in the 90s, it was realized that for one-dimensional models with positive Lyapunov exponents some parts of multi-scale analysis can be replaced by considerations involving subharmonicity and large deviation estimates for the corresponding cocycle, leading to nonperturbative proofs for 1D quasiperiodic models. In this paper we present a short proof along these lines, for the Anderson model. To prove dynamical localization we also develop a uniform version of Craig-Simon's bound that works in high generality and may be of independent interest.

math-ph↗
arXiv2018

Cantor spectrum of graphene in magnetic fields

We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux $Φ/2π$ through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero.

math.SP↗
arXiv2018

Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase

We prove sharp spectral transition in the arithmetics of phase between localization and singular continuous spectrum for Diophantine almost Mathieu operators. We also determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices throughout the localization region. This uncovers a universal structure in their behavior governed by the exponential phase resonances. The structure features a new type of hierarchy, where self-similarity holds upon alternating reflections.

math-ph↗
arXiv2017

Universal hierarchical structure of quasiperiodic eigenfunctions

We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the corresponding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied.

math-ph↗
arXiv2017

Discrete Bethe-Sommerfeld Conjecture

In this paper, we prove a discrete version of the Bethe-Sommerfeld conjecture. Namely, we show that the spectra of multi-dimensional discrete periodic Schrödinger operators on $\mathbb{Z}^d$ lattice with sufficiently small potentials contain at most two intervals. Moreover, the spectrum is a single interval, provided one of the periods is odd, and can have a gap whenever all periods are even.

math-ph↗
arXiv2017

Complex one-frequency cocycles

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 setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other hand, this last property provides a good control of the periodic approximations of a cocycle, allowing us to show that domination can be characterized, in the presence of a gap in the Lyapunov spectrum, by additional regularity of the dependence of sums of Lyapunov exponents.

math.DS↗
arXiv2016

Arithmetic Spectral Transitions for the Maryland Model

We give a precise description of spectra of the Maryland model $ (h_{λ,α,θ}u)_n=u_{n+1}+u_{n-1}+ λ\tan π(θ+nα)u_n$ for all values of parameters. We introduce an arithmetically defined index $δ(α, θ)$ and show that for $α\notin\mathbb{Q},\,$ $σ_{sc}(h_{λ,α,θ})=\overline{\{e:γ_λ(e) <δ(α, θ) \}}$ and $σ_{pp}(h_{λ,α,θ})=\{e:γ_λ(e) \geq δ(α, θ) \}$. Since $σ_{ac}(h_{λ,α,θ})=\emptyset,\;$ this gives complete description of the spectral decomposition for {\it all} values of parameters $λ,α,θ$, making it the first case of a family where arithmetic spectral transition is described without any parameter exclusion. The set of eigenvalues can be explicitly identified for all parameters, using the {\it quantization condition}. We also establish, for the first time for this or any other model, a quantization condition for singular continuous spectrum (an arithmetically defined measure zero set that supports singular continuous measures) for all parameters.

math-ph↗
arXiv2016

Second Phase transition line

We study the phase transion line of the almost Mathieu operator, that separates arithmetic regions corresponding to singular continuous and a.e. pure point regimes, and prove that both purely singular continuous and a.e. pure point spectrum occur for dense sets of frequencies.

math-ph↗
arXiv2016

Quantum dynamical bounds for ergodic potentials with underlying dynamics of zero topological entropy

In this paper we obtain upper quantum dynamical bounds as a corollary of positive Lyapunov exponent for Schrödinger operators $H_{f,θ} u(n)=u(n+1)+u(n-1)+ ϕ(f^nθ)u(n)$, where $ϕ: \mathcal{M}\to {\Bbb R}$ is a piecewise Hölder function on a compact Riemannian manifold $\mathcal{M}$, and $f:\mathcal{M}\to\mathcal{M}$ is a uniquely ergodic volume preserving map with zero topological entropy. As corollaries we obtain localization-type statements for shifts and skew-shifts on higher dimensional tori with arithmetic conditions on the parameters. These are the first localization-type results with precise arithmetic conditions for multi-frequency quasiperiodic and skew-shift potentials.

math.SP↗
arXiv2016

Singular continuous spectrum for singular potential

We prove that Schrödinger operators with meromorphic potentials $(H_{α,θ}u)_n=u_{n+1}+u_{n-1}+ \frac{g(θ+nα)}{f(θ+nα)} u_n$ have purely singular continuous spectrum on the set $\{E: L(E)<δ{(α,θ)}\}$, where $δ$ is an explicit function, and $L$ is the Lyapunov exponent. This extends results of Jitomirskaya and Liu for the Maryland model and of Avila, You and Zhou for the almost Mathieu operator, to the general family of meromorphic potentials.

math.SP↗