伪黎曼双曲空间的共振与留数算子
Resonances and residue operators for pseudo-Riemannian hyperbolic spaces
AI总结：
本文研究实、复、四元数与八元数伪黎曼双曲空间上 Laplace–Beltrami 算子预解式的亚纯延拓，显式确定全部共振，并将留数像刻画为退化主序列的子表示。
摘要：
对于任意定义在 $\mathbb{R}$、$\mathbb{C}$、$\mathbb{H}$ 或 $\mathbb{O}$ 上的伪黎曼双曲空间 $X$，我们证明 $X$ 上 Laplace–Beltrami 算子 $-\Box$ 的预解式 $R(z)=(\Box-z\operatorname{Id})^{-1}$ 可作为一族从 $C_c^\infty(X)$ 到 $\mathcal{D}'(X)$ 的算子，亚纯延拓穿过 $\Box$ 的谱。其极点称为共振，我们在所有情形下显式确定了这些共振。对于每个共振，相应留数算子在 $\mathcal{D}'(X)$ 中的像构成 $X$ 的等距群的一个表示，我们将其识别为退化主序列的一个子表示。我们的研究特别包括 de Sitter 空间和 Anti-de Sitter 空间上的偶函数情形。对于黎曼对称空间，Miatello–Will 与 Hilgert–Pasquale 已得到类似结果。黎曼与非黎曼情形之间的主要定性差异在于：对于非黎曼空间，预解式可具有二阶极点，并且可在 $R(z)$ 所延拓到的覆盖的分支点处具有极点，同时留数表示可以是无限维的。
英文摘要：
For any pseudo-Riemannian hyperbolic space $X$ over $\mathbb{R},\mathbb{C},\mathbb{H}$ or $\mathbb{O}$, we show that the resolvent $R(z)=(\Box-z\operatorname{Id})^{-1}$ of the Laplace-Beltrami operator $-\Box$ on $X$ can be extended meromorphically across the spectrum of $\Box$ as a family of operators $C_c^\infty(X)\to \mathcal{D}'(X)$. Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in $\mathcal{D}'(X)$ forms a representation of the isometry group of $X$, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces. For Riemannian symmetric spaces analogous results were obtained by Miatello-Will and Hilgert-Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which $R(z)$ extends, and the residue representations can be infinite-dimensional.
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