Resonances and residue operators for symmetric spaces of rank one
Resonances and residue operators for symmetric spaces of rank one
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一阶对称空间的共振和留数算子
DOI：
10.1016/j.matpur.2009.01.009
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
A. Pasquale
中科院分区：
文献类型：
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作者：
J. Hilgert;A. Pasquale
设X=G/K是非紧型秩一黎曼对称空间，−Δ是X上的Laplace-Beltrami算子。我们证明了Δ的预解算子R（z）在谱上是亚纯连续的，并且明确地确定了极点，即共振点.进一步，我们用G的有限维球面表示来描述剩余算子。结果回答了M. Zworski在[M. Zworski，What are the residues of the resolvent of the Laplacian on non-compact symmetric spaces？2006年，在莱森堡施洛斯的2006年IRTG暑期学校举办的研讨会。见http：//math.berkeley.edu/reisensburg.pdf]。剩余算子的秩是从限制根版本的Weyl维数公式的球形最高权重表示，我们证明了任意对称空间的非紧型。
Let X=G/K be a rank-one Riemannian symmetric space of the noncompact type and let −Δ be the Laplace–Beltrami operator on X. We show that the resolvent operator R(z) of Δ can be meromorphically continued across the spectrum and explicitly determine the poles, i.e. the resonances. Further we describe the residue operators in terms of finite-dimensional spherical representations of G. The result answers a question posed by M. Zworski in [M. Zworski, What are the residues of the resolvent of the Laplacian on non-compact symmetric spaces? Seminar held at the IRTG Summer School 2006, Schloss Reisensburg, 2006. Available at http://math.berkeley.edu/~zworski/reisensburg.pdf]. The rank of the residue operators is derived from a restricted root version of the Weyl dimension formula for spherical highest weight representations which we prove for arbitrary symmetric spaces of the noncompact type.