刘小会^1,2, 曹建文¹, 张娅¹

1. 中国科学院软件研究所, 北京 100190;
2. 中国科学院大学, 北京 100190

收稿日期:2017-01-22出版日期:2018-03-15发布日期:2018-02-03


基金资助:国家自然科学基金~（91430214，91230109，11301507）资助项目.

THE ISO-SPECTRAL PROBLEM, REVISITED IN PLANAR CASE

Liu Xiaohui^1,2, Cao Jianwen¹, Zhang Ya¹

1. Institute of Software, Chinese Academy of Science, Beijing 100190, China;
2. University of Chinese Academy of Science, Beijing 100190, China

Received:2017-01-22Online:2018-03-15Published:2018-02-03







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听音辨鼓这个反问题发展至今已经半个世纪，许多数学和物理学家都做出了很多有益的贡献.这个挑战性问题由美国数学家M.Kac 1966年正式提出，用数学语言描述为欧几里得空间中，是否可以找到两个（或更多）非等距单连通区域是等谱的?C.Gordon等人1992年在二维平面上给出一对等谱区域，首次对Kac的问题说"No".问题发展至今，只有17类平面等谱区域.它们都遵循一系列镜像反演规则，成对等谱，保持反演规则不变，改变基本构建块的形状，可以形成无穷多同类的等谱对.本文重访17类等谱区域，探究构建块之间的镜像反演规则.通过折叠方法，建立17类等谱区域特征函数之间的迁移映射关系.结合符号计算，列出17类等谱区域移植矩阵的通解.此外，利用Bernstein-Bézier多项式，计算等谱区域的广义特征值.
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[1]	孙家昶, 张娅. 二维等谱问题研究的计算数学框架[J]. 计算数学, 2017, 39(3): 229-286.

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