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 $\hat{I}{»}_{\ast }$. 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 $(\hat{I}{»}_{\ast },q_0)$, liberating the theory from the prevailing constraint $\hat{I}{»}_{\ast }>\hat{I}{»}_1(q_0)$ (where $\hat{I}{»}_{\ast }$ is the observed eigenvalue and $\hat{I}{»}_1$ 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 $\Vert \hat{q}-q_0{\Vert }_{{\mathcal{L}}^p}$ 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.

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