Entire functions and compact operators with $S_p$-imaginary component

紧致 Anosov 流形上阻尼波的谱集中现象

The spectral concentration for damped waves on compact Anosov manifolds

本文研究紧致 Anosov 流形上阻尼波的特征值分布，利用 Anosov 测地流的中偏差原理证明大多数特征值虚部以对数速度集中于阻尼函数平均值附近，并将其应用于扭曲 Selberg zeta 函数零点的集中性。

An example of resonance instability

We construct a semiclassical Schrödinger operator such that the imaginary part of its resonances closest to the real axis changes by a term of size $h$ when a real compactly supported potential of size $o ( h )$ is added.

Spectral gaps and abelian covers of convex co-compact surfaces

Given a convex co-compact hyperbolic surface $X=Γ\backslash \mathbb{H}^2$, we investigate the resonance spectrum $\mathcal{R}_j$ of the laplacian $Δ_j$ on large finite abelian covers $X=Γ_j\backslash \mathbb{H}^2$, where $Γ_j$ is a finite index normal subgroup of $Γ$. Let $δ$ be the Hausdorff dimension of the limit set of $Γ$. We show that there exists an $\varepsilon>0$, such that for all $j$, resonances $\mathcal{R}_j$ in $\{ δ-\varepsilon< \mathrm{Re}(s) \leq δ\}$ are all real and satisfy a Weyl law given by the degree of the cover i.e. $\vert Γ/ Γ_j\vert$. In particular, we prove that for …

Regularized Laplacian determinants of self-similar fractals

We study the spectral zeta functions of the Laplacian on fractal sets which are locally self-similar fractafolds, in the sense of Strichartz. These functions are known to meromorphically extend to the entire complex plane, and the locations of their poles, sometimes referred to as complex dimensions, are of special interest. We give examples of self-similar sets such that their complex dimensions are not on the imaginary axis, which allows us to interpret their Laplacian determinant as the regularized product of their eigenvalues. We then investigate a connection between the logarithm of the d…