论文标题
台球的动力Zeta功能
Dynamical zeta functions for billiards
论文作者
论文摘要
令$ d \ subset {\ mathbb r}^d，\：d \ geqslant 2，$为有限的成对分离的有限集合严格凸出紧凑型障碍。令$μ_j\ in {\ Mathbb C}，\：{\ rm im} \：μ_j> 0，$是$ d $的laplacian的共振，neumann或noumann或dirichlet dirichlet dirichlet dirichlet dirichlet边界条件在$ \ partial d $上。对于$ d $奇数，$ u（t）= \ sum_j e^{i | t | μ_j} $是$ \ Mathcal {d}'（{\ Mathbb r} \ setMinus \ {0 \}）$的分布，$ U（t）$的领先奇点的拉普拉斯变换产生了动态Zeta Zeta Zeta Zeta函数$η__ {\ Mathrm n} $ n} $ n} $ n}，\： Neumann和Dirichlet边界条件。这些ZETA功能在分析共振的分布中起着至关重要的作用。在非复制条件（1.1）下，对于$ d \ geqslant 2 $，我们表明$η_{\ mathrm n} $和$η_\ Mathrm d $允许在整个复杂平面上持续meromorthic延续。在边界$ \ partial d $的特定情况下，使用Fried（1995）的结果，我们证明函数$η_\ mathrm {d} $不能全部。遵循Ikawa（1988）的结果，这意味着存在{\ Mathbb c}中的条$ \ {z \ {z \ in {\ mathbb c}：\：0 <{\ rm im} \：z \leqα\} $包含无限的共振数量$μ_jj$，以解决dirichlet的问题。此外，对于$α\ gg 1 $，我们获得了该带中的共振的下限。
Let $D \subset {\mathbb R}^d,\: d \geqslant 2,$ be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let $μ_j \in {\mathbb C},\: {\rm Im}\: μ_j > 0,$ be the resonances of the Laplacian in the exterior of $D$ with Neumann or Dirichlet boundary condition on $\partial D$. For $d$ odd, $u(t) = \sum_j e^{i |t| μ_j}$ is a distribution in $ \mathcal{D}'({\mathbb R} \setminus \{0\})$ and the Laplace transforms of the leading singularities of $u(t)$ yield the dynamical zeta functions $η_{\mathrm N},\: η_{\mathrm D}$ for Neumann and Dirichlet boundary conditions, respectively. These zeta functions play a crucial role in the analysis of the distribution of the resonances. Under the non-eclipse condition (1.1), for $d \geqslant 2$ we show that $η_{\mathrm N}$ and $η_\mathrm D$ admit a meromorphic continuation to the whole complex plane. In the particular case when the boundary $\partial D$ is real analytic, by using a result of Fried (1995), we prove that the function $η_\mathrm{D}$ cannot be entire. Following the result of Ikawa (1988), this implies the existence of a strip $\{z \in {\mathbb C}: \: 0 < {\rm Im}\: z \leqα\}$ containing an infinite number of resonances $μ_j$ for the Dirichlet problem. Moreover, for $α\gg 1$ we obtain a lower bound for the resonances lying in this strip.
