The Selberg Zeta Function for Convex Co-Compact Schottky Groups
The Selberg Zeta Function for Convex Co-Compact Schottky Groups
复制标题
凸共紧肖特基群的 Selberg Zeta 函数
DOI：
10.1007/s00220-003-1007-1
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发表时间：
2002
影响因子：
2.4
通讯作者：
M. Zworski
中科院分区：
文献类型：
--
作者：
Laurent Guillopé;Kevin K. Lin;M. Zworski
本文给出了双曲空间n+1上的凸余紧Schottky群的Selberg zeta函数的一个新的上界：在平行于虚轴的带中，zeta函数由exp（C）有界|S|其中δ是群的极限集的维数。该界比最优全局界exp（C| S| n+1），并给出了Γ\n+1的共振（散射极点）个数的新界.这个结果的证明是基于应用全纯L2-技术的Ruelle转移算子的行列式的研究和准自相似的极限集。我们还研究了这个问题的数值，并提供证据表明，界可能是最佳的。我们的动机来自于分子动力学，我们认为Γ n+1是量子混沌散射的最简单的模型。
We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on the hyperbolic space ℍn+1: in strips parallel to the imaginary axis the zeta function is bounded by exp (C|s|δ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp (C|s|n+1) , and it gives new bounds on the number of resonances (scattering poles) of Γ\ℍn+1 . The proof of this result is based on the application of holomorphic L2-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider Γ\ℍn+1 as the simplest model of quantum chaotic scattering.