谱隙中eigenvalue计数函数的算子理论方法

通过傅里叶衰减计算双曲曲面的共振与谱隙

Resonance counting and spectral gaps for hyperbolic surfaces via Fourier decay

该研究针对无限面积凸余紧双曲曲面，利用极限集的豪斯多夫维数与Patterson–Sullivan测度的傅里叶衰减指数，证明了改进的分形魏尔上界，进而得到其本质谱隙的更尖锐上界。

齐次度量树上的薛定谔算子：间隙谱

Schrödinger operator on homogeneous metric trees: spectrum in gaps

本文研究了齐次根度量树上的Schr dinger算子$A_{gV} = A_0 + gV$的谱性质，该算子具有衰减的实位势$V$和耦合常数$g\ge 0$ .自由拉普拉斯算子$A_0 = -Δ$的谱具有带隙结构，在每个有限间隙的中间具有无限重数的单个本征值。微扰$gV$在带隙中产生额外的本征值。如果势$V$具有固定的符号，则这些特征值是$g$的单调函数。假设后一个条件得到满足，$V$是对称的，即取决于到树根的距离，我们进行了详细的渐近分析的计数函数的离散特征值的极限$g\to\infty$。根据$V$的符号和衰变，这种渐近性要么是外尔型的，要么完全由$V$在无穷远处的行为决定。

周期矩阵值Jacobi算子的李雅普诺夫函数

Lyapunov functions for periodic matrix-valued Jacobi operators

我们考虑周期矩阵值Jacobi算子。这个算子的谱是绝对连续的，并且由间隔隔开的区间组成。我们定义的李雅普诺夫函数，这是分析相关的黎曼曲面。在每一张上，李雅普诺夫函数具有标量情形下李雅普诺夫函数的标准性质。我们表明，这个功能有（真正的或复杂的）分支点，我们称之为共振。我们证明了存在两种类型的间隙：i）稳定间隙，即，端点是周期和反周期本征值，ii）不稳定（谐振）间隙，即，端点是谐振（实分支点）。我们表明，一些谱数据确定的Jacobi算子的谱（计数多重性）。

Spectral gaps and discrete magnetic Laplacians

The aim of this article is to give a simple geometric condition that guarantees the existence of spectral gaps of the discrete Laplacian on periodic graphs. For proving this, we analyse the discrete magnetic Laplacian (DML) on the finite quotient and interpret the vector potential as a Floquet parameter. We develop a procedure of virtualising edges and vertices that produces matrices whose eigenvalues (written in ascending order and counting multiplicities) specify the bracketing intervals where the spectrum of the Laplacian is localised. We prove Higuchi-Shirai's conjecture for Z-periodic tre…

本征值分布在本质谱间隙中的Bochner-薛定谔算子

Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator

本文研究了在具有有界几何的流形上，Bochner-薛定谔算子在本质谱间隙中的本征值分布，并推导了与这些本征值相关的迹渐进行式。

Asymptotics of spectral gaps of quasi-periodic Schrödinger operators

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…