Eigenvalues associated with a closed geodesic
Eigenvalues associated with a closed geodesic
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与闭合测地线相关的特征值
DOI：
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发表时间：
1976
期刊：
影响因子：
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通讯作者：
A. Weinstein
中科院分区：
文献类型：
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作者：
V. Guillemin;A. Weinstein
1.背景。来自量子力学和光学的直观论证表明，紧致黎曼流形X上的闭测地线(周期“粒子”)和Laplace-Beltrami算子的本征值(周期“波”)之间应该存在某种联系。事实上，1959年，Huber[8]证明了对于X为常负曲率曲面，X上的闭测地线的长度集合和Ax的谱彼此决定。Huber给出的这两个数字序列之间的关系非常复杂，以至于在给定一个序列的情况下，要明确地找到另一个序列是极其困难的。最近，Colin de Verdière[2]，然后是Chazarain[1]以及Duistermaat和Guillmin[3]已经证明，对于任何可微流形上的大多数黎曼度量，拉普拉斯谱决定闭地线的长度及其模4的Morse指数。这里，闭地线的长度表现为分布的奇点
1. Background. Intuitive arguments drawn from quantum mechanics and optics suggest that there should be some relation between the closed geodesies (periodic "particles") on a compact Riemannian manifold X and the eigenvalues (periodic "waves") of the Laplace-Beltrami operator A^. Indeed, in 1959, Huber [8] proved, for X a surface of constant negative curvature, that the set of lengths of closed geodesies on X and the spectrum of Ax determine one another. The relation given by Huber between these two sequences of numbers is sufficiently complicated to make it extremely difficult to find one sequence explicitly, given the other. Recently, Colin de Verdière [2], then Chazarain [1] and Duistermaat and Guillemin [3] have shown that, for most Riemannian metrics on any differentiable manifold, the spectrum of the Laplacian determines the lengths of the closed geodesies and their Morse indices modulo 4. Here, the lengths of the closed geodesies appear as the singular points of the distribution