Mathematical Sciences: Spectral Problems and Inverse Spectral Problems

本课题主要研究三个方面的问题 数学分析的相关领域。 在每种情况下， 继续进行先前类似性质的调查。 第一个目标来自于努力应用逆 自伴常微分的散射结果 运营商 这些操作符在分析中自然出现， 已知可积的非线性波动方程。 虽然可以通过逆方法获得显式解， 解决方案的长期行为还没有得到很好的理解， 还远远没有完成。 困难在于 群速度并不独立于能量而保持固定， 所以扰动是双向的 这意味着 自由解不是完整解的好近似， 相关线性问题的散射解。 一个好 一般初始数据问题的参数为 寻找。 工作也将进行矩阵求逆问题。 与微分方程的联系是通过最近的 发现自伴矩阵的对角化可以 在可积哈密顿系统的背景下。 这是 进一步观察到，QR，LU和Cholesky算法， 非对称矩阵也适用于这种情况。 而且 这种方法的计算潜力是最有前途的。 将继续研究的长期渐近的工作， 非对称矩阵的Rayleigh移位QR算法。 第三个关注的主题涉及光谱问题， 薛定谔算子。 一个重要的问题出现在 晶体着色理论围绕着 扰动薛定谔算子的特征值。 更准确地说， 人们感兴趣的是确定特征值 一个受扰动的运营商可以实现在间隙中， 原始操作符的频谱。 这种现象被认为是 发生，问题是要找到合适的信息， 频谱分布

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