Inverse Spectral Problem for Analytic Domains I: Balian-Bloch Trace Formula
Inverse Spectral Problem for Analytic Domains I: Balian-Bloch Trace Formula
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解析域的逆谱问题 I：Balian-Bloch 迹公式
DOI：
10.1007/s00220-004-1074-y
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发表时间：
2001
影响因子：
2.4
通讯作者：
S. Zelditch
中科院分区：
文献类型：
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作者：
S. Zelditch
这是第一次在一系列文件[Z3，Z4]反谱/共振问题的解析平面域Ω。本文给出了Balian-Bloch迹公式的严格形式[BB 1，BB 2].它是当k→∞且τ>0时Ω的Dirichlet或Neumann Laplacian的正则化预解式的迹Tr 1 ΩRρ（k+iτ log k）的渐近公式。当的支集恰好包含一条周期反射射线γ的长度Lγ时，则Tr 1 ΩRρ（k+iτ log k）的渐近展开式与γ处的波迹展开式基本相同.这种方法存在的理由是它导致了相对简单的波不变量的显式公式。例如，我们给出了平面域上弹跳球轨道的波不变量的第一个公式（细节将在[Z3]中出现）。虽然我们只在维度2中提供细节，但方法和结果扩展到所有维度，几乎没有修改。
This is the first in a series of papers [Z3, Z4] on inverse spectral/resonance problems for analytic plane domains Ω. In this paper, we present a rigorous version of the Balian-Bloch trace formula [BB1, BB2]. It is an asymptotic formula for the trace Tr1ΩRρ(k+iτ log k) of the regularized resolvent of the Dirichlet or Neumann Laplacian of Ω as k→∞ with τ>0. When the support of contains the length Lγ of precisely one periodic reflecting ray γ, then the asymptotic expansion of Tr1ΩRρ(k+iτ log k) is essentially the same as the wave trace expansion at γ. The raison d’ètre for this approach is that it leads to relatively simple explicit formulae for wave invariants. For example, we give the first formulae for wave invariants of bouncing ball orbits of plane domains (the details will appear in [Z3]). Although we only present details in dimension 2, the methods and results extend with few modifications to all dimensions.