论文标题
频谱完成和反向Sturm-liouville问题
Spectrum completion and inverse Sturm-Liouville problems
论文作者
论文摘要
给出了方程式-y {\ prime} {\ prime}+q（x）y =λy的常规sturm -liouville问题的有限特征值，其电位q（x）是未知的。我们显示了计算更多特征值的可能性，而无需有关潜在Q（x）的任何其他信息。 Moreover, considering the Sturm-Liouville problem with the boundary conditions y{\prime}(0)-hy(0)=0 and y{\prime}(π)+Hy(π)=0, where h, H are some constants, we complete its spectrum without additional information neither on the potential q(x) nor on the constants h and H. The eigenvalues are computed with a uniform absolute accuracy.基于此结果，我们提出了一种新方法，用于从两个光谱中恢复电势的逆sturm-liouville问题。该方法包括第一步中光谱的完成，并将其还原为第二个线性代数方程系统。从溶液向量的第一个组件中恢复了电势Q（x）。该方法基于特殊的Neumann系列Bessel功能表示，用于具有具有显着特性的Sturm-Liouville方程解决方案，并导致有效的数值算法来解决逆Sturm-Liouville问题。
Given a finite set of eigenvalues of a regular Sturm-Liouville problem for the equation -y{\prime}{\prime}+q(x)y=λy, the potential q(x) of which is unknown. We show the possibility to compute more eigenvalues without any additional information on the potential q(x). Moreover, considering the Sturm-Liouville problem with the boundary conditions y{\prime}(0)-hy(0)=0 and y{\prime}(π)+Hy(π)=0, where h, H are some constants, we complete its spectrum without additional information neither on the potential q(x) nor on the constants h and H. The eigenvalues are computed with a uniform absolute accuracy. Based on this result we propose a new method for numerical solution of the inverse Sturm-Liouville problem of recovering the potential from two spectra. The method includes the completion of the spectra in the first step and reduction to a system of linear algebraic equations in the second. The potential q(x) is recovered from the first component of the solution vector. The approach is based on special Neumann series of Bessel functions representations for solutions of Sturm-Liouville equations possessing remarkable properties and leads to an efficient numerical algorithm for solving inverse Sturm-Liouville problems.
