Quantitative theory of the inverse spectral problem for Sturm-Liouville operator with applications

An interesting inverse optimization spectral problem, with important applications in structural health monitoring and damage detection, material design, seismic wave analysis, sonar detection, and related fields, involves reconstructing a potential $\hat{q}$ from a finite set of observed eigenvalues such that $\hat{q}$ yields an optimal approximation of the target potential $q_0$. Previous efforts have been confined to qualitative analysis, whereas the quantitative counterpart remains an open problem. This paper introduces a quantitative framework for the inverse spectral problem by using a phase plane analysis (planar dynamical system approach). We provide a quantitative characterization of the relationship between the reconstructed potential $\hat{q}$, its target potential $q_0$, and the observed eigenvalue $\lambda_*$. Remarkably, for ${q} \in \mathcal{L}^2$, our analysis yields a substantially stronger conclusion: an exact analytical expression for the reconstructed potential $\hat{q}$. In other words, our framework yields a complete resolution of the optimization inverse spectral problem in the $\mathcal{L}^2$ case. Moreover, we establish the uniqueness of $\hat{q}$ {\bf for any $q_0, \lambda_*\in \mathbb R$}, a key advance that eliminates the need for traditional constraints linking $\lambda_*$ and $q_0$. An additional finding is the construction of a homeomorphic mapping that reveals the dilation relation between the errors $\|\hat{q} - q_0\|_{\mathcal L^p}$ associated with the $m$-th eigenvalue and the principal eigenvalue. A summary of the main results, along with practical applications in engineering and mathematical physics, concludes this work.