具有延迟参数的Sturm-Liouville算子的谱特征及其反问题

Spectral Characterizations for Sturm-Liouville Operators with a Constant Delay and Their Inverse Spectral Problems

具有延迟参数的Sturm-Liouville算子的逆谱问题是Sturm-Liouville理论的一个重要分支.延迟参数的Sturm-Liouville微分方程可用于对声波信号的传输以及液压冲击或其他波过程的模拟.本文主要研究有限区间上具有延迟变量的二阶Sturm-Liouville算子的特征值及其逆谱问题,利用势函数的Fourier展开式的方法得出了相应的逆谱问题的唯一性定理,即在某些假设条件下由边值问题的两组谱可以唯一确定延迟变量τ和势函数q(x).最后本文给出势函数的重构算法.

The inverse spectral problem of Sturm-Liouville operators with a constant delay is an important branch of Sturm-Liouville theory.The Sturm-Liouville differential equation with a constant delay can be used to model the transmission of acoustic signals,hydraulic shock or other wave processes.In this paper,the authors mainly study the eigenvalues and inverse spectrum problems of second-order Sturm-Liouville operators with a constant delay on finite interval.And the authors also obtain the uniqueness theorems by Fourier expansion of the potential function.That is,under certain assumptions,the delay variableτand the potential function can be uniquely determined by the two sets of spectra of the boundary value problem.Finally the authors give the reconstruction algorithms of the potential function.