Florin P. Boca, Alexandru Zaharescu
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.
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We prove that the exponential moments of the position operator stay bounded for the supercritical almost Mathieu operator with Diophantine frequency.
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 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 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 thickn…
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…
Twenty dry Martinis for the Unitary Almost Mathieu Operator
本文在非临界区域中，针对具有丢番图频率的酉几乎 Mathieu 算子，解决了干十马提尼问题。
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.
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 $c'q_n^{-\varkappa/2}$, $c'>0$.
We consider point spectrum traces in the Hofstadter model. We show how to recover the full quantum Hofstadter trace by integrating these point spectrum traces with the appropriate free density of states on the lattice. This construction is then generalized to the almost Mathieu operator and its n-th moments which can be expressed in terms of generalized Kreft coefficients.
Plotting spectra of a range of almost Mathieu operators reveals a beautiful fractal-like image that contains multiple copies of a butterfly image. We demonstrate that plotting the butterflies using a gap-labelling scheme based on K-theory or Chern numbers reveals systematic discontinuities in the gap positioning. A proper image is produced only when we take into account these discontinuities, and close the butterfly wingtips at the points of discontinuity. A conjecture is presented showing a simple formula for locating the discontinuities, and numerical evidence is given to support the conject…
The Sharp Spectral Transition for Almost Mathieu Operators via Alternating Resonances
该研究通过交替共振方法，证明了几乎Mathieu算子在特定条件下呈现Anderson局域化，解决了相关文献中的猜想。
We determine numerically the self-similarity maps for spectra of the almost Mathieu operators, a two-dimensional fractal-like structure known as the Hofstadter butterfly. The similarity maps each have a horizontal component determined by certain algebraic maps, and vertical component determined by a Mobius transformation, indexed by a semigroup of the matrix group $GL_2(\Z)$. Based on the numerical evidence, we state and prove a continuity result for the similarity maps. We note a connection between the indexing of the similarity maps and Morita equivalence of rotation algebras $A_θ$, a contin…
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…
Log-Hausdorff multifractality of the absolutely continuous spectral measure of the almost Mathieu operator
本文研究了亚临界几乎Mathieu算子绝对连续谱测度的分形特征，通过连分数与度量Diophantine技巧给出了经典多重分形谱的完整描述及对数尺度下的精细描述，严格证实了物理学家的直觉。
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…
Arithmetic spectral transition for the unitary almost Mathieu operator
本文研究了一维准周期单位ary算子的算术谱过渡，证明了在特定频率条件下实现安德森局部化，扩展了之前关于Diophantine频率的局部化结果。
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…
In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \[ (H_{b,ϕ} x)_n= x_{n+1} +x_{n-1} + b \cos(2 πn ω+ ϕ)x_n \] on $l^2(\mathbb{Z})$ and its associated eigenvalue equation to deduce that for $b \ne 0,\pm 2$ and $ω$ Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ``Ten Martini Problem'' for these values of $b$ and $ω$. Moreover, we prove that for $|b|\ne 0$ small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open.
Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum
本文通过将临界几乎Mathieu族表示为奇异Jacobi矩阵，证明了其谱的Hausdorff维数不超过1/2，解决了长期问题，并推广了谱测度连续性与量子图石墨烯的维数结果。
Cantor Spectrum via a Reducibility-Duality Bridge for the Mosaic Almost Mathieu Operator
针对镶嵌准Mathieu算子，通过建立对偶框架并将态密度积分与纤维旋转数的对应关系推广到该模型，结合Aubry对偶、可约性理论和Moser-Pöschel论证，证明了所有非临界参数下谱为Cantor集。
Golden mean renormalization for the almost Mathieu operator and related skew products
本文研究圆旋转上的 SL(2,R) 斜积，证明黄金均值重整化具有长度 3 的非平凡周期轨道，并用数值结果说明其与 Hofstadter 蝴蝶谱端及广义特征函数标度性质的联系。
Almost Everything About the Unitary Almost Mathieu Operator
本研究引入源自均匀磁场二维量子行走的酉近马蒂厄算子，证明其Aubry–André对偶性，刻画各参数区转移矩阵上闭链动力学，精确计算谱Lyapunov指数并明确不同参数下的谱类型，细化了临界情形下的谱性质结果。
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 consider skew-product maps over circle rotations $x\mapsto x+α$ (mod 1) with factors that take values in SL(2,R). This includes maps of almost Mathieu type. In numerical experiments, with $α$ the inverse golden mean, Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic scaling behavior that is reminiscent of critical phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior indeed occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormal…
Generalized Aubry-André formula and continuity of the intersection spectrum of the Almost Mathieu operator
本文研究 Almost Mathieu 运算符的谱，证明了 Lebesgue 测度在交集谱上的矩是耦合常数 λ 的多项式，系数为频率 α 的三角多项式，推广了 Aubry-André 公式，并证明了测度在参数下的连续性及分析性。
Kicked Harper operators and on-resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectrums of these operators. In this paper we apply C*-algebra methods to explain this resemblance. We show that each pair of corresponding operators belong to a common rotation C*-algebra B_α, prove that their spectrums are equal if αis irrational, and prove that the Hausdorff distance between their spectrums converges to zero as q increases if α= p/q with p and q coprime inte…
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 develop some non-perturbative methods for studying the IDS in almost Mathieu and related models. Assuming positive Lyapunov exponents, and assuming that the potential function is a trigonometric polynomial of degree k, we show that the Holder exponent of the IDS is almost 1/2k. We also show that this is stable under small perturbations of the potential (e.g., potentials which are close to that of almost Mathieu again give rise to almost 1/2 Holder continuous IDS). Moreover, off a set of Hausdorff dimension zero the IDS is Lipschitz. We further deduce from these properties that the IDS is ab…
We consider a system of fermions with a quasi-random almost-Mathieu disorder interacting through a many-body short range potential. We establish exponential decay of the zero temperature correlations, indicating localization of the interacting ground state, for weak hopping and interaction and almost everywhere in the frequency and phase; this extends the analysis in \cite{M} to chemical potentials outside spectral gaps. The proof is based on Renormalization Group and is inspired by techniques developed to deal with KAM Lindstedt series.
The Dry Ten Martini Problem at Criticality
本文证明几乎Mathieu算子在临界耦合下每个允许能隙标签均由开放能隙实现，通过最大Lyapunov指数单调性及边界平衡方法确立临界能隙开启。
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…
For a class of discrete quasi-periodic Schroedinger operators defined by covariant re- presentations of the rotation algebra, a lower bound on phase-averaged transport in terms of the multifractal dimensions of the density of states is proven. This result is established under a Diophantine condition on the incommensuration parameter. The relevant class of operators is distinguished by invariance with respect to symmetry automorphisms of the rotation algebra. It includes the critical Harper (almost-Mathieu) operator. As a by-product, a new solution of the frame problem associated with Weyl-Heis…
Sharp Spectral Gaps, Arithmetic Localization, and Reducibility via Resonance Analys
本文通过共振分析，研究了准周期Schrödinger算子的谱间隙、算术定位及可约性问题，解决了多个开放性猜想并揭示了亚临界算子的分层结构。
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