We consider separable 2D discrete Schrödinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J. Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to separable multidimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine. The proof is based on the study of the thickness of the spectrum of the almost Mathieu operator, and utilizes the Newhouse Gap Lemma on sums of Cantor sets.
We derive a new Chambers-type formula and prove sharper upper bounds on the measure of the spectrum of critical almost Mathieu operators with rational frequencies.
We prove that the exponential decay rate in expectation is well defined and is equal to the Lyapunov exponent, for supercritical almost Mathieu operators with Diophantine frequencies.
We show that there exists a dense set of frequencies with positive Hausdorff dimension for which the Hausdorff dimension of the spectrum of the critical almost Mathieu operator is positive.
For the almost Mathieu operator with a small coupling constant, for a series of spectral gaps, we describe the asymptotic locations of the gaps and get lower bounds for their lengths. The results are obtained by analysing a monodromy matrix.
We study the almost Mathieu operator at critical coupling. We prove that there exists a dense $G_δ$ set of frequencies for which the spectrum is of zero Hausdorff dimension.
We prove that for Diophantine \om and almost every þ, the almost Mathieu operator, (H_{ω,λ,θ}Ψ)(n)=Ψ(n+1) + Ψ(n-1) + λ\cos 2π(ωn +θ)Ψ(n), exhibits localization for λ> 2 and purely absolutely continuous spectrum for λ< 2. This completes the proof of (a correct version of) the Aubry-André conjecture.
We introduce self-similar versions of the one-dimensional almost Mathieu operators. Our definition is based on a class of self-similar Laplacians instead of the standard discrete Laplacian, and includes the classical almost Mathieu operators as a particular case. Our main result establishes that the spectra of these self-similar almost Mathieu operators can be completely described by the spectra of the corresponding self-similar Laplacians through the spectral decimation framework used in the context of spectral analysis on fractals. In addition, the self-similar structure of our model provide…
We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon's list of Schrödinger operator problems for the twenty-first century.
We estimate the norm of the almost Mathieu operator $H_{θ,λ} =U+U^*+(λ/2)(V+V^*)$ in the rotation $C^*$-algebra $A_θ=C^*(U,V unitaries;UV=e^{2πiθ} VU)$. In this process, we significantly improve the inequality $\| H_θ\| \leq 2\sqrt{2}$, $θ\in [0.25,0.5]$, conjectured by Beguin, Valette and Zuk.
We consider the spectrum of the almost Mathieu operator $H_α$ with frequency $α$ and in the case of the critical coupling. Let an irrational $α$ be such that $|α-p_n/q_n|<c q_n^{-\varkappa}$, where $p_n/q_n$, $n=1,2,\dots$ are the convergents to $α$, and $c$, $\varkappa$ are positive absolute constants, $\varkappa<56$. Assuming certain conditions on the parity of the coefficients of the continued fraction of $α$, we show that the central gaps of $H_{p_n/q_n}$, $n=1,2,\dots$, are inherited as spectral gaps of $H_α$ of length at least $c'q_n^{-\varkappa/2}$, $c'>0$.
It is known that the spectral type of the almost Mathieu operator depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya's conjecture in \cite{Ji95,J07}. Together with \cite{Aab}, we give the sharp description of phase transitions for the almost Mathieu operator.
We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon's list of Schrödinger operator problems for the twenty-first century.
Let $α\in \mathbb{R}\backslash \mathbb{Q}$ and $β(α) = \limsup _{n \to \infty}(\ln q_{n+1})/ q_n <\infty$, where $p_n/q_n$ is the continued fraction approximations to $α$. Let $(H_{λ,α,θ}u) (n)=u(n+1)+u(n-1)+ 2λ\cos2π(θ+nα)u(n)$ be the almost Mathieu operator on $\ell^2(\mathbb{Z})$, where $λ, θ\in \mathbb{R}$. Avila and Jitomirskaya \cite{avila2009ten} conjectured that for $2θ\in α\mathbb{Z} + \mathbb{Z}$, $H_{λ,α,θ}$ satisfies Anderson localization if $|λ|>e^{2β(α)}$. In this paper, we developed a method to treat simultaneous frequency and phase resonances and obtain that for $2θ\in α\mathbb…
We show that some spectral properties of the almost Mathieu operator with frequency well approximated by rationals can be as poor as at all possible in the class of all one-dimensional discrete Schroedinger operators. For the class of critical coupling, we show that the Hausdorff measure of the spectrum may vanish (for appropriately chosen frequencies) whenever the gauge function tends to zero faster than logarithmically. For arbitrary coupling, we show that modulus of continuity of the integrated density of states can be arbitrary close to logarithmic; we also prove a similar result for the L…
We discuss spectral characteristics of a one-dimensional quantum walk whose coins are distributed quasi-periodically. The unitary update rule of this quantum walk shares many spectral characteristics with the critical Almost-Mathieu Operator; however, it possesses a feature not present in the Almost-Mathieu Operator, namely singularity of the associated cocycles (this feature is, however, present in the so-called Extended Harper's Model). We show that this operator has empty absolutely continuous spectrum and that the Lyapunov exponent vanishes on the spectrum; hence, this model exhibits Canto…
For the almost Mathieu operator $(H_{λ,α,θ}u)_n=u_{n+1}+u_{n-1}+2λ\cos2π(θ+nα)u_n$, Avila and Jitomirskaya guess that for a.e. $θ$, $H_{λ,α,θ}$ satisfies Anderson localization if $ |λ| > e^{ β} $, and they establish this for $ |λ| > e^{\frac{16}{9} β}$. In the present paper, we extend their result to regime $ |λ| > e^{\frac{3}{2} β}$.
These notes are based on a series of six lectures, given during my stay at the CRC 701 in June/July 2008. The lecture series intended to give a survey of some of the results for the almost Mathieu operator that have been obtained since the early 1980's. Specifically, the metal-insulator transition is discussed in detail, along with its relation to the ten Martini problem via duality and reducibility.
For almost Mathieu operator $(H_{λ,α,θ}u)_n=u_{n+1}+u_{n-1}+2λ\cos2π(θ+nα)u_n$, the dry version of Ten Martini problem predicts that the spectrum $Σ_{λ,α}$ of $ H_{λ,α,θ}$ has all gaps open for all $λ\neq 0$ and $ α\in \mathbb{R}\backslash \mathbb{Q}$. Avila and Jitomirskaya prove that $Σ_{λ,α}$ has all gaps open for Diophantine $α$ and $0<|λ|<1$. In the present paper, we show that $Σ_{λ,α}$ has all gaps open for all $ α\in \mathbb{R}\backslash \mathbb{Q}$ with small $λ$.
The almost Mathieu operator is the discrete Schrödinger operator $H_{α,β,θ}$ on $\ell^2(\mathbb{Z})$ defined via $(H_{α,β,θ}f)(k) = f(k + 1) + f(k - 1) + β\cos(2παk + θ) f(k)$. We derive explicit estimates for the eigenvalues at the edge of the spectrum of the finite-dimensional almost Mathieu operator. We furthermore show that the (properly rescaled) $m$-th Hermite function $ϕ_m$ is an approximate eigenvector of this operator, and that it satisfies the same properties that characterize the true eigenvector associated to the $m$-th largest eigenvalue. Moreover, a properly translated and modula…
We initiate an approach to simultaneously treat numerators and denominators of Green's functions arising from quasi-periodic Schrödinger operators, which in particular allows us to study completely resonant phases of the almost Mathieu operator. Let $ (H_{λ,α,θ}u) (n)=u(n+1)+u(n-1)+ 2λ\cos2π(θ+nα)u(n)$ be the almost Mathieu operator on $\ell^2(\mathbb{Z})$, where $λ, α, θ\in \mathbb{R}$. Let $$ β(α)=\limsup_{k\rightarrow \infty}-\frac{\ln ||kα||_{\mathbb{R}/\mathbb{Z}}}{|k|}.$$ We prove that for any $θ$ with $2θ\in α\mathbb{Z}+\mathbb{Z}$, $H_{λ,α,θ}$ satisfies Anderson localization if $|λ|>e^…
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. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrodinger operator, and allows us to complete the proof of the Aubry-Andre conjecture on the measure of the spectrum of the Almost Mathieu Operator.
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.
We introduce a notion of $β$-almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded $β$-almost periodic potentials. Applications include a sharp arithmetic criterion of full spectral dimensionality for analytic quasiperiodic Schrödinger operators in the positive Lyapunov exponent regime and arithmetic criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator.
We establish sharp results on the modulus of continuity of the distribution of the spectral measure for one-frequency Schrodinger operators with Diophantine frequencies in the region of absolutely continuous spectrum. More precisely, we establish 1/2-Holder continuity near almost reducible energies (an essential support of absolutely continuous spectrum). For non-perturbatively small potentials (and for the almost Mathieu operator with subcritical coupling), our results apply for all energies.
We generalize the approach to localization in one dimension introduced by Kunz-Souillard, and refined by Delyon-Kunz-Souillard and Simon, in the early 1980's in such a way that certain correlations are allowed. Several applications of this generalized Kunz-Souillard method to almost periodic Schrödinger operators are presented. On the one hand, we show that the Schrödinger operators on $l^2(\mathbb{Z})$ with limit-periodic potential that have pure point spectrum form a dense subset in the space of all limit-periodic Schrödinger operators on $l^2(\mathbb{Z})$. More generally, for any bounded po…
We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with $δ$ coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the field grows linearly with an irrational slope, measured in terms of the flux through the loops of the chain, we demonstrate the character of the spectrum relating it to the almost Mathieu operator.
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 Schödinger operators with (measure-theoretically) typical quasi-periodic analytic potentials and fixed strong Diophantine frequency. As applications, we show the discrete version of Deift's 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…
We study discrete Schroedinger operators $(H_{α,θ}ψ)(n)= ψ(n-1)+ψ(n+1)+f(αn+θ)ψ(n)$ on $l^2(Z)$, where $f(x)$ is a real analytic periodic function of period 1. We prove a general theorem relating the measure of the spectrum of $H_{α,θ}$ to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of $H_{α,θ}$ are positive. For the almost Mathieu operator ($f(x)=2λ\cos 2πx$) it follows that the measure of the spectrum is equal to $4|1-|λ||$ for all real $θ$, $λ\ne\pm 1$, and all irrational $α$.
We consider the one-dimensional discrete Schrödinger operator $$ \bigl[H(x,ω)φ\bigr](n)\equiv -φ(n-1)-φ(n+1) + V(x + nω)φ(n)\ , $$ $n \in \mathbb{Z}$, $x,ω\in [0, 1]$ with real-analytic potential $V(x)$. Assume $L(E,ω)>0$ for all $E$. Let $\mathcal{S}_ω$ be the spectrum of $H(x,ω)$. For all $ω$ obeying the Diophantine condition $ω\in \mathbb{T}_{c,a}$, we show the following: if $\mathcal{S}_ω\cap (E',E")\neq \emptyset$, then $\mathcal{S}_ω\cap (E',E")$ is homogeneous in the sense of Carleson (see [Car83]). Furthermore, we prove, that if $G_i$, $i=1,2$ are two gaps with $1 > |G_1| \ge |G_2|$, t…
