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The spectral concentration for damped waves on compact Anosov manifolds

Yulin Gong

TL;DR

This work analyzes the high-frequency spectrum of the damped wave equation on compact Anosov manifolds, showing that most eigenvalues lie in a logarithmically narrow region near the average damping $-\overline{a}$ as $\Re \tau_n\to\infty$, with width $w(h)=|\log h|^{-{\frac{1-\alpha}{2}}}$ for any $0<\alpha<1$. The authors develop a semiclassical reduction and averaging scheme, then employ a moderate deviation principle for the Anosov geodesic flow to obtain quantitative concentration bounds, together with precise resolvent estimates for a perturbed damped operator. A key consequence is that nontrivial zeros of twisted Selberg zeta functions concentrate in a logarithmic region approaching $\Re s=\tfrac{1}{2}$ as $|\Im s|\to\infty$, linking spectral data of non-self-adjoint operators to dynamical properties of the geodesic flow. The results advance the understanding of resonance distributions in damped, non-self-adjoint systems and connect quantum-chaotic techniques with zeta-function phenomenology in hyperbolic geometry.

Abstract

We study the spectral distribution of damped waves on compact Anosov manifolds. Sjöstrand \cite{SJ1} proved that the imaginary parts of the majority of the eigenvalues concentrate near the average of the damping function, see also Anantharaman \cite{AN2}. In this paper, we prove that the most of eigenvalues actually lie in certain regions with imaginary parts that approach the average logarithmically as the real parts tend to infinity. The proof relies on the moderate deviation principles for Anosov geodesic flows. As an application, we show the concentration of non-trivial zeros of twisted Selberg zeta functions in a logarithmic region asymptotically close to $\Re s=\frac{1}{2}$.

The spectral concentration for damped waves on compact Anosov manifolds

TL;DR

This work analyzes the high-frequency spectrum of the damped wave equation on compact Anosov manifolds, showing that most eigenvalues lie in a logarithmically narrow region near the average damping as , with width for any . The authors develop a semiclassical reduction and averaging scheme, then employ a moderate deviation principle for the Anosov geodesic flow to obtain quantitative concentration bounds, together with precise resolvent estimates for a perturbed damped operator. A key consequence is that nontrivial zeros of twisted Selberg zeta functions concentrate in a logarithmic region approaching as , linking spectral data of non-self-adjoint operators to dynamical properties of the geodesic flow. The results advance the understanding of resonance distributions in damped, non-self-adjoint systems and connect quantum-chaotic techniques with zeta-function phenomenology in hyperbolic geometry.

Abstract

We study the spectral distribution of damped waves on compact Anosov manifolds. Sjöstrand \cite{SJ1} proved that the imaginary parts of the majority of the eigenvalues concentrate near the average of the damping function, see also Anantharaman \cite{AN2}. In this paper, we prove that the most of eigenvalues actually lie in certain regions with imaginary parts that approach the average logarithmically as the real parts tend to infinity. The proof relies on the moderate deviation principles for Anosov geodesic flows. As an application, we show the concentration of non-trivial zeros of twisted Selberg zeta functions in a logarithmic region asymptotically close to .

Paper Structure

This paper contains 11 sections, 15 theorems, 139 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Semiclassical reduction
  3. Application to Zeros of the Twisted Selberg Zeta Function
  4. Related work
  5. Preliminary
  6. Anosov Geodesic Flow on Compact Manifolds
  7. Semiclassical Microlocal Analysis
  8. Averaging and Perturbations
  9. Control the Perturbation
  10. Proof of main Theorems \ref{['mainthm1']}, \ref{['concentrationforDWE']}, and \ref{['zerosoftsz']}
  11. Appendix: Proof of moderate deviation principles Theorem \ref{['MDPA']}

Key Result

Theorem 1.1

Assume that $M$ is a $d$-dimensional Anosov compact manifold, $a \in C^{\infty}(M, \mathbb{R})$, and $\{\tau_n\}$ is the spectrum of the operator DWEop. Let $\overline{a} = \frac{1}{\mathrm{Vol}(M)}\int_{M}a(x)\,d\mathrm{Vol}(x)$ be the average of the damping function $a(x)$ on $M$, there exists a c

Figures (3)

  • Figure 1: Spectral concentration region: The red line marks the average of the damping term, the blue line marks the boundary of a logarithmic region asymptotically approaching the average, and the green line marks the boundary of a horizontal strip around the average.
  • Figure 2: The horizontal axis represents $q^{T}(x,\xi)$, and the vertical axis represents $p(x,\xi)$. The colored region represents the energy shell $p^{-1}\left(\left[\frac{1}{2}-\epsilon_{0},\frac{1}{2}+\epsilon_{0}\right]\right)$, where $q^{T} = \langle q \rangle_{T}$ in this region. The yellow region represents the part where $\langle q \rangle_{T} \leq \overline{q} + \kappa w(h)$, and in this region, $q^{T} = \chi_{h}(q^{T})$. The blue region represents the part where $\langle q \rangle_{T} \geq \overline{q} + \kappa w(h)$, and this region contains $\mathrm{supp}\left(\widehat{q}^{T}\right)$.
  • Figure 3: The blue line is the boundary of domain $\Omega_{h}^{1}$, the green line is the smooth boundary of domain $\Omega_{0}$, the black line is the boundary of domain $\Omega_{0}^{\prime}$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2: Markus and Matsaev MAMV, Sjöstrand SJ1
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Moderate deviations
  • Corollary 2.4
  • Remark
  • Definition 2.5
  • ...and 13 more