Mathematical Sciences: Spectral Problems and Inverse Spectral Problems

这个项目集中在三个数学分析领域：(1)超椭圆黎曼曲面和哈密顿力学的周期；(2)矩阵问题；(3)可积系统的长时间行为。第一个主题是研究算术几何平均值与超椭圆积分计算和哈密顿力学之间的新联系。第二个主题是分析计算矩阵奇异值的高精度算法。本课题的这一部分也涉及矩阵乘积的特征值和特征向量的计算问题。在分析中起中心作用的是可积哈密顿系统，它在整数时间内插值奇异值/特征值算法的迭代。第三个主题涉及可积系统的长期行为分析，特别是户田震问题。注意将集中在低激波速度的情况下，更高激波速度的更一般的情况下，平衡状态下双无限Toda晶格局部摄动的稳定性问题，特征值填充“幻影”带的速率和方式的问题，以及Gel’fand- dikii层次中非线性波动方程的长时间行为分析。本课题的工作涉及光谱问题和逆光谱问题。该项目涉及哈密顿力学的应用，改进计算矩阵奇异值的准确性的算法的发展，以及可积系统的长期行为的研究。

This project is focused on three areas of mathematical analysis: (1) periods of hyperelliptic Riemann surfaces and Hamiltonian mechanics, (2) matrix problems, and (3) long-time behavior of integrable systems. The first topic involves the investigation of a new connection between the arithmetic- geometric mean and the computation of hyperelliptic integrals on the one hand, and Hamiltonian mechanics on the other. The second topic involves the analysis of high accuracy algorithms to compute the singular values of a matrix. This part of the project is also concerned with the problem of computing the eigenvalues and eigenvectors of products of matrices. A central role in the analysis is played by integrable Hamiltonian systems which interpolate the iterates of the singular value / eigenvalue algorithms at integer times. The third topic involves an analysis of the long-time behavior of integrable systems, specifically of the Toda shock problem. Attention will be focused on the case of low shock speed, the more general case for higher shock speeds, the stability problem in which a doubly infinite Toda lattice in equilibrium is perturbed locally, the question of the rate and the manner in which the eigenvalues fill the "phantom" band, and the analysis of the long-time behavior of the nonlinear wave equations in the Gel'fand-Dikii hierarchy. The work in this project deals with spectral problems and inverse spectral problems. The project is concerned with applications to Hamiltonian mechanics, with the development of algorithms which improve the accuracy of the computed singular values of a matrix, and with the study of the long-time behavior of integrable systems.