A dynamical system framework yielding quantitative inverse spectral results for Sturm-Liouville operators

Abstract

This paper establishes a dynamical-system framework that yields quantitative results for the inverse optimal spectral problem of reconstructing a potential $\hat{q}$ from finite observed eigenvalues to achieve an optimal approximation of the target potential $q_0$. Previous efforts relying on convex analysis have been confined solely to {\em qualitative} analysis due to the inherent limitations of convex-analytic techniques for inverse problems, while the {\bf quantitative} counterpart has remained an open problem. Based on our dynamical-system framework, we provide a quantitative characterization of the relationship between the reconstructed potential $\hat{q}$, its target potential $q_0$, and the observed eigenvalue $\lambda_*$. In particular, for ${q} \in \mathcal{L}^2$, our framework yields a substantially stronger conclusion. Remarkably, our dynamical-system framework secures the uniqueness of $\hat{q}$ over the full parameter space $(\lambda_*, q_0)$, liberating the theory from the prevailing constraint $\lambda_* > \lambda_1(q_0)$ (where $\lambda_*$ is the observed eigenvalue and $\lambda_1$ is the principle eigenvalue). This stands in sharp contrast to classical approaches, which rely heavily on convex-set analysis and are inherently confined by its stringent assumptions. 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 structural health monitoring and damage detection, material design, seismic wave analysis, sonar detection, and related fields, concludes this work.