Partial Information for Inverse Spectral Uniqueness in Vibration System with Multiple Frozen Arguments
Lung-Hui Chen
TL;DR
This work addresses the inverse spectral problem for a Sturm–Liouville system with multiple frozen arguments, seeking $q(x)$ from the spectrum of a boundary-value problem that includes $N$ fixed observation points. It develops integral identities linking the difference of two potentials to the frozen points via the four characteristic functions and employs zero-density theory (via Titchmarsh and Levin) to assess uniqueness under partial information about $q$ and the set $\{a_i\}$. The main contributions are three conditional uniqueness results: uniqueness or non-uniqueness depends on the zero-density behavior of $\sum_{i=1}^N\cos{\rho a_i}$ at specific spectral sequences, and on how $\hat{q}$ is supported relative to the frozen points, with irrational independence of $\{a_i/\pi\}$ playing a key role. These findings provide theoretical criteria for reconstructability with partial measurements and inform sensor placement strategies in vibration systems.
Abstract
In this paper, we investigate the inverse spectral problem of the Sturm-Liouville operator with many frozen arguments fixed at the points $\{a_{1}, a_{2},\ldots,a_{N}\}$ in $(0,π)$. We start with counting the zeros or the eigenvalues of characteristic function, and then discuss how certain information provided a priori on the point set $\{a_{1}, a_{2},\ldots,a_{N}\}$ would affect the uniqueness or non-uniqueness of this vibration system with many frozen points. The knowledge at the frozen or regulator points are practical in many on-site problems. Parallelly, certain irrational independence assumption assures the inverse spectral uniqueness as well.