Mathematical Sciences: Spectral Problems and Inverse Spectral Problems

本项目重点研究数学分析的三个相关领域。在每一种情况下，这项工作都是在继续早期类似性质的调查。第一个目标源于对自伴随常微分算子应用逆散射结果的努力。这些算符在分析已知可积的非线性波动方程时自然出现。虽然可以通过逆方法得到显式解，但解的长时间行为还不清楚，而且还远远不够完整。困难在于群速度的符号，群速度不受能量的影响而保持固定，所以扰动向两个方向移动。这意味着对于相关线性问题的全散射解，自由解不是很好的近似。对于具有一般初始数据的问题，将寻求一个好的参数。工作也将在矩阵反演问题上进行。与微分方程的联系是通过最近的发现，自伴随矩阵的对角化可以在可积哈密顿系统的背景下看待。进一步观察到，非对称矩阵的QR、LU和Cholesky算法也适用于这种情况。此外，这种方法的计算潜力是最有希望的。工作将继续研究非对称矩阵的瑞利移位QR算法的长期渐近性。第三个关注的主题是薛定谔算子的谱问题。晶体着色理论中出现的一个重要问题是围绕摄动薛定谔算子的特征值的存在性。更确切地说，人们感兴趣的是确定一个扰动算子的特征值在多大程度上可以在原始算子的频谱间隙内实现。这种现象是已知发生的，问题是找到合适的光谱分布信息

This project focuses on research concerned with three related areas of mathematical analysis. In each case the work continues earlier investigations of a similar nature. The first objectives derive from efforts to apply inverse scattering results for selfadjoint ordinary differential operators. These operators arise naturally in the analysis of nonlinear wave equations which are known to be integrable. Although explicit solutions can be obtained by inverse methods, the long time behavior of solutions is not well understood, and is far from complete. The difficulty lies in the sign of the group velocity which does not stay fixed independent of energy, so disturbances move out in both directions. This means that free solutions are not good approximations for the full scattering solutions of the associated linear problem. A good parametrix for the problem with general initial data will be sought. Work will also be carried out on matrix inversion problems. The connection with differential equations is made through recent discoveries that diagonalization of self-adjoint matrices can be viewed in the context of integrable Hamiltonian systems. It was further observed that the QR, LU and Cholesky algorithms for nonsymmetric matrices also fit into this context. Moreover, the computational potential for this approach is most promising. Work will continue in studying the long term asymptotics of the Rayleigh shifted QR algorithm for nonsymmetric matrices. A third topic of concern deals with spectral problems for Schrodinger operators. An important question arising in the theory of crystal coloration revolves around the existence of eigenvalues of a perturbed Schrodinger operator. More precisely, one is interested in determining the extent to which eigenvalues of a perturbed operator can materialize within gaps in the spectrum of the original operator. The phenomenon is known to occur, the issue is to find suitable information on the distribution of the spectrum