椭圆闭测地线上的波不变量
Wave invariants at elliptic closed geodesics
摘要：
本文研究紧致黎曼流形（M，g）上拉普拉斯算子的谱不变量，即波不变量.若U（t）表示（M，g）的波群，则当t = 0或ti是一条闭测地线的长度时，迹Tr U（t）是奇异的.它在每个长度处都有一种特殊类型的奇异性展开，其系数被称为波不变量。我们的主要目的是计算当闭测地线为非退化椭圆时，以曲率、Jacobi场等表示的波不变量。我们通过在闭测地线处将拉普拉斯算子转换为量子Birkhoff标准形来实现这一点。这种正规形式以前由V.Guillemin介绍过。我们给出了一个新的算法来计算它，并表示波不变量的正规形不变量。
英文摘要：
This paper concerns spectral invariants of the Laplacian on a compact Riemannian manifold (M,g) known as wave invariants. If U(t) denotes the wave group of (M,g), then the trace Tr U(t) is singular when t = 0 or when ti is the length of a closed geodesic. It has a special type of singularity expansion at each length and the coefficients are known as the wave invariants. Our main purpose is to calculate the wave invariants explicitly in terms of curvature, Jacobi fields etc. when the closed geodesic is non-degenerate elliptic. We do this by putting the Laplacian into quantum Birkhoff normal form at the closed geodesic. Such a normal form was previously introduced by V. Guillemin. We give a new algorithm for calculating it, and for expressing wave invariants in terms of normal form invariants.
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