Partial Information for Inverse Spectral Uniqueness in Vibration System with Multiple Frozen Arguments

TL;DR

This paper explores the inverse spectral problem for Sturm-Liouville operators with multiple fixed points, analyzing how partial information and irrational independence influence the uniqueness of the vibration system reconstruction.

Contribution

It introduces conditions under which inverse spectral uniqueness holds or fails for systems with multiple frozen points, considering partial data and irrational independence assumptions.

Findings

Partial information affects spectral uniqueness.

Irrational independence ensures inverse spectral uniqueness.

Conditions for non-uniqueness are identified.

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 $(0,\pi)$. 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.