Plancherel-P\'{o}lya's Type of Instability in Vibration System with Multiple Frozen Arguments

TL;DR

This paper investigates the inverse spectral problem for Sturm-Liouville operators with multiple fixed points, analyzing spectral perturbations using Plancherel-Pólya inequalities and connecting them to potential function perturbations in L² norm.

Contribution

It introduces a novel approach to spectral perturbation analysis in Sturm-Liouville problems with multiple frozen arguments using Plancherel-Pólya inequalities.

Findings

Spectral perturbations can be characterized by Plancherel-Pólya type inequalities.

Perturbations relate to changes in potential functions within L² space.

The approach links spectral data perturbations to potential function variations.

Abstract

We discuss the problem of the inverse spectral problem of Sturm-Liouville operator with multiple frozen arguments at $\{a_{1}, a_{2},\ldots,a_{N}\}$ in $(0,\pi)$. One may consider the characteristic functions as perturbation of sine or of cosine functions depending on the boundary problem prescribed. However, such perturbation is represented in the form of Fourier transform of certain function which may or may not bring in Riesz basis theory and classical perturbation theory in functional analysis. We shall demonstrate the spectral perturbation in Plancherel-P\'{o}lya's type of inequality and connect to perturbation of related potential functions in $L^{2}$-functional norm.