Alexander Fedotov
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.
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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 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.
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 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…
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.
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.
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。
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 $λ$.
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…
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 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…
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。
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。
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。
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.
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^…
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。
High-Index Asymptotics of Mathieu Spectral Gaps and Splittings
该研究推导了马蒂厄算子在高指数n下的谱隙与分裂的渐近公式，补充了前人未得到的高阶相对修正项，并通过等谱性将结果推广到另一类势函数。
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。
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…
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。
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.
The Hill operator admits a band gap structure. As a special case, like the Mathieu operator, one has only open gaps, however, the instability intervals of the Whittaker-Hill operator may be open or closed. In 2007, P. Djakov and B. Mityagin gave the asymptotics of band gaps for a special Whittaker-Hill operator [P. Djakov and B. Mityagin, J. Funct. Anal., 242, 157-194 (2007).]. In this paper, a more general Whittaker-Hill operator is considered and the asymptotics of the instability intervals are studied.
Spectral regularity and defects for the Kohmoto model
本文研究Kohmoto模型及其蝴蝶谱，利用Farey数证明谱估计，并确定有理旋转处导致谱缺陷的杂质，结果与Almost-Mathieu算子及Hofstadter蝴蝶的情形类似。
Spectra of Non-Self-Adjoint Almost Mathieu Matrices and the Scottish Flag Operator
本文研究非自伴几乎马蒂厄矩阵的谱，推导相关钱伯斯公式，分离特征多项式的特定部分，证明其零点分布性质，计算偶数阶矩阵的极限特征值测度，明确苏格兰旗矩阵的谱位置。
We establish an asymptotic formula, uniformly down to the Planck scale, for the number of small gaps between the first N eigenvalues of the Laplacian on almost all flat tori and also on almost all rectangular flat tori.
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.
It is shown that the complete localization of eigenvectors for the almost Mathieu operator entails the absence of Cantor spectrum for this operator.
We consider $C=A+B$ where $A$ is selfadjoint with a gap $(a,b)$ in its spectrum and $B$ is (relatively) compact. We prove a general result allowing $B$ of indefinite sign and apply it to obtain a $(δV)^{d/2}$ bound for perturbations of suitable periodic Schrodinger operators and a (not quite)Lieb-Thirring bound for perturbations of algebro-geometric almost periodic Jacobi matrices.
Exact local distribution of the absolutely continuous spectral measure
该研究拓展了单频薛定谔算子谱测度连续性的相关结论，精确刻画了稠密小势场下谱测度的局部分布，并探究了其在次临界能量下的分层赫尔德连续性。
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…
Monotonicity, global symplectification and the stability of Dry Ten Martini Problem
本文研究了在超临界 regime 下，对于固定无理频率和三角多项式势，证明了满足间隙标记条件的I型能量边界是开放光谱间隙的边界，并展示了几乎马蒂厄算子在小扰动下的鲁棒性。
For non-self-adjoint almost-periodic Schrödinger operators, a criterion is given to guarantee that they have both the same spectrum and same Lyapunov exponents with the discrete free Laplacian. As a byproduct, we show that the Moser-Pöschel argument for opening gaps may not be valid for non-self-adjoint operators.
Square-Root Profiles and Sharp Fourier Decay of Quasiperiodic Spectral Measures
该研究针对一维准周期薛定谔算子，证明了次临界算子的尖锐可测集界、傅里叶衰减等结果，建立了几乎Mathieu算子的谱密度平方根包络，通过无反射测度定理及Borel变换残差展开解决了相关问题并证实了猜想。
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 Schrödinger operators with small Gevrey potentials, the length of spectral gaps decays sub-exponentially with respect to its labelling, the long range duality operator has pure point spectrum with sub-exponentially decaying eigenfunctions for almost all phases and the spectrum is an interval for discrete Schrödinger operator acting on $ \mathbb{Z}^d $ with small separable potentials. All these …
The spectrum of a Schrödinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m, we consider the problem of maximizing the gap-to-midgap ratio for the m-th spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the m-th gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly m interv…
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