论文标题
椭圆的征函数和光谱刚度
Eigenfunction asymptotics and spectral Rigidity of the ellipse
论文作者
论文摘要
本文是有关椭圆的同一问题的系列的一部分。在本文中，我们研究了椭圆形或诺伊曼边界条件的椭圆征算数据的库奇数据。使用许多在椭圆本征函数上的经典结果，我们确定了本本征函数库奇数据的微局部缺陷度量。椭圆有可集成的台球，即边界阶段空间是由台球地图的不变曲线叶子的。我们证明，对于任何不变的曲线$ c $，都存在一系列本征函数，其cauchy数据集中在$ c $上。我们使用此结果给出了一个新的证据，表明椭圆在$ c^{\ infty} $域中与椭圆形的对称性相对的频谱固定。
This paper is part of a series concerning the isospectral problem for an ellipse. In this paper, we study Cauchy data of eigenfunctions of the ellipse with Dirichlet or Neumann boundary conditions. Using many classical results on ellipse eigenfunctions, we determine the microlocal defect measures of the Cauchy data of the eigenfunctions. The ellipse has integrable billiards, i.e. the boundary phase space is foliated by invariant curves of the billiard map. We prove that, for any invariant curve $C$, there exists a sequence of eigenfunctions whose Cauchy data concentrates on $C$. We use this result to give a new proof that ellipses are infinitesimally spectrally rigid among $C^{\infty}$ domains with the symmetries of the ellipse.
