Analytic continuation of resolvent kernels on noncompact symmetric spaces
Analytic continuation of resolvent kernels on noncompact symmetric spaces
复制标题
非紧对称空间上解析核的解析延拓
DOI：
10.1007/s00209-004-0760-y
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发表时间：
2003
影响因子：
0.8
通讯作者：
A. Strohmaier
中科院分区：
文献类型：
--
作者：
A. Strohmaier
设X=G/K是非紧型对称空间，Δ是X上与G-不变度量相关的拉普拉斯算子。我们证明了Δ的预解核允许黎曼曲面的全纯扩张依赖于对称空间的秩。这个黎曼曲面是复平面的一个分支覆盖，其中去掉了真实的轴的某一部分。它在Δ的光谱底部有一个分支点。它进一步表明，这个分支点是二次的，如果X的秩是奇数，否则是对数。当G只有一个共轭类的Cartan子代数时，预解核扩展到G的分支覆盖上的全纯函数，唯一的分支点是谱的底。
Abstract.Let X=G/K be a symmetric space of noncompact type and let Δ be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of Δ admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane with a certain part of the real axis removed. It has a branching point at the bottom of the spectrum of Δ. It is further shown that this branching point is quadratic if the rank of X is odd, and is logarithmic otherwise. In case G has only one conjugacy class of Cartan subalgebras the resolvent kernel extends to a holomorphic function on a branched cover of ℂ with the only branching point being the bottom of the spectrum.