The inverse spectral problem for the discrete cubic string

Two inverse spectral problems for a class of singular Krein strings

We solve the inverse problem from the spectral measure and the inverse three-spectra problem for the class of singular Krein strings on a finite interval with trace class resolvents. In particular, this includes a complete description of all possible spectral measures and three (Dirichlet) spectra associated with this class of Krein strings. The solutions of these inverse problems are obtained by approximation with Stieltjes strings.

An inverse spectral problem for a star graph of Krein strings

We solve an inverse spectral problem for a star graph of Krein strings, where the known spectral data comprises the spectrum associated with the whole graph, the spectra associated with the individual edges as well as so-called coupling matrices. In particular, we show that these spectral quantities uniquely determine the weight within the class of Borel measures on the graph, which give rise to trace class resolvents. Furthermore, we obtain a concise characterization of all possible spectral data for this class of weights.

The inverse spectral problem for indefinite strings

Motivated by the study of certain nonlinear wave equations (in particular, the Camassa-Holm equation), we introduce a new class of generalized indefinite strings associated with differential equations of the form \[-u"=z\,u\,ω+z^2u\,\upsilon\] on an interval $[0,L)$, where $ω$ is a real-valued distribution in $H^{-1}_{\mathrm{loc}}[0,L)$, $\upsilon$ is a non-negative Borel measure on $[0,L)$ and $z$ is a complex spectral parameter. Apart from developing basic spectral theory for these kinds of spectral problems, our main result is an indefinite analogue of M. G. Krein's celebrated solution of …

Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings

We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems …

有限雅可比矩阵与Krein-Stieltjes弦的反问题

Inverse problems for finite Jacobi matrices and Krein--Stieltjes strings

本文研究有限雅可比矩阵和Krein-Stieltjes弦的动态反问题，提出三种基于动态Dirichlet-to-Neumann算子恢复未知参数的方法，并给出动态反演数据的特征刻画。

关于弹性弦树状网络逆问题

On an inverse problem for tree-like networks of elastic strings

本文研究树状网络弹性弦的逆问题，采用波动方程边界控制方法，通过叶节点测量恢复弦的光学长度及树拓扑结构。