临界几乎Mathieu算子：隐藏奇异性、谱隙连续性与谱的Hausdorff维数

广义 Aubry-André 公式与 Almost Mathieu 运算符交集谱的连续性

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。

Continuity of the measure of the spectrum for discrete quasiperiodic operators

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 $α$.

On the regularity of the Hausdorff distance between spectra of perturbed magnetic Hamiltonians

We study the regularity properties of the Hausdorff distance between spectra of continuous Harper-like operators. As a special case we obtain Hölder continuity of this Hausdorff distance with respect to the intensity of the magnetic field for a large class of magnetic elliptic (pseudo)differential operators with long range magnetic fields.

Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators

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.

Holder continuity of absolutely continuous spectral measures for one-frequency Schrodinger operators

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.

绝对连续谱测度的精确局部分布

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。

Continuity of nonlinear eigenvalues in $CD(K,\infty)$ spaces with respect to measured Gromov-Hausdorff convergence

In this note we prove in the nonlinear setting of $CD(K,\infty)$ spaces the stability of the Krasnoselskii spectrum of the Laplace operator $-Δ$ under measured Gromov-Hausdorff convergence, under an additional compactness assumption satisfied, for instance, by sequences of $CD^*(K,N)$ metric measure spaces with uniformly bounded diameter. Additionally, we show that every element $λ$ in the Krasnoselskii spectrum is indeed an eigenvalue, namely there exists a nontrivial $u$ satisfying the eigenvalue equation $- Δu = λu$.

Log-dimensional bounds for the spectral measure of the disordered Holstein model

We demonstrate Hausdorff log dimensional bounds on the continuity of the spectral measure of the strongly disordered Holstein operator.

多维准周期Schrödinger算子的Cantor谱

Cantor spectrum for multidimensional quasi-periodic Schrödinger operators

研究多维准周期Schrödinger算子的Cantor谱问题，证明临界近正交算子的谱Hausdorff维数可任意小但非零。

Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D

The existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schrödinger operators. In a forthcoming work [9] (arXiv:1709.00975) this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions cove…

关于某些一维 Schrödinger 算子的分形连续性性质

On Fractal Continuity Properties of Certain One-Dimensional Schrödinger Operators

本文构造一维 Schrödinger 算子示例，揭示全直线与半直线谱测度在 packing 维数、Hausdorff 维数及 Borel 集测度上的反常分形连续性。

动态定义与顺从算子族的谱估计

Spectral estimates of dynamically-defined and amenable operator families

本文研究了动力系统定义的核算子，证明了在群作用和核满足不同连续性条件时谱映射的 Hölder 与 Lipschitz 连续性，推广并连接了关于准周期与子位移 Schrödinger 算子的已有结果，且不限于整数格或阿贝尔群。

Spectral Continuity for Aperiodic Quantum Systems I. General Theory

How does the spectrum of a Schrödinger operator vary if the corresponding geometry and dynamics change? Is it possible to define approximations of the spectrum of such operators by defining approximations of the underlying structures? In this work a positive answer is provided using the rather general setting of groupoid $C^\ast$-algebras. A characterization of the convergence of the spectra by the convergence of the underlying structures is proved. In order to do so, the concept of continuous field of groupoids is slightly extended by adding continuous fields of cocycles. With this at hand, m…

局部化区域中态密度的积分绝对连续性

Absolute continuity of the integrated density of states in the localized regime

本文证明了大三角势准周期薛定谔算子积分态密度的绝对连续性，部分解决Eliasson开放问题，并推广至长程算子，基于分层定量几乎可约性，证实其2017年猜想。

On the spacing of zeros of paraorthogonal polynomials for singular measures

We prove a lower bound on the spacing of zeros of paraorthogonal polynomials on the unit circle, based on continuity of the underlying measure as measured by Hausdorff dimensions. We complement this with the analog of the result from arXiv:1011.3159 showing that clock spacing holds even for certain singular continuous measures.