等谱形变和谱刚性的不变量
Invariants of isospectral deformations and spectral rigidity
摘要：
我们引入了连续变形的弱等谱性概念。考虑具有Robin边界条件的紧致黎曼流形上的Laplace-Beltrami算子。给定旋转向量满足丢番图条件的台球映射的Kronecker不变环面$Λ$，证明了$Λ$上涉及Robin边界条件下函数的某些积分在弱等谱形变下保持常数.为此，我们构造连续的家庭准模与$Λ$。我们还获得了等谱不变量的拉普拉斯算子与实值潜在的连续变形的潜在的紧流形上。作为一个应用程序，我们证明了频谱刚性的情况下，刘维台球表的二维。
英文摘要：
We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace-Beltrami operator on a compact Riemannian manifold with boundary with Robin boundary conditions. Given a Kronecker invariant torus $Λ$ of the billiard ball map with a vector of rotation satisfying a Diophantine condition we prove that certain integrals on $Λ$ involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes associated with $Λ$. We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. As an application we prove spectral rigidity in the case of Liouville billiard tables of dimension two.
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