Application of approximation theory by nonlinear manifolds in Sturm-Liouville inverse problems

TL;DR

This paper explores limitations and improvements in reconstructing Sturm-Liouville potentials using approximation theory, establishing negative bounds and proposing near-optimal formulas for eigenvalue estimation.

Contribution

It provides new negative results on approximation limits and introduces an almost optimal reconstruction formula for Sturm-Liouville inverse problems.

Findings

Negative approximation bounds of order $(\omega\ln\omega)^{-(m+1)}$

Eigenvalue and characteristic value estimations for Sturm-Liouville operators

An almost optimal reconstruction formula of order $\omega^{-m}$

Abstract

We give here some negative results in Sturm-Liouville inverse theory, meaning that we cannot approach any of the potentials with $m+1$ integrable derivatives on $\mathbb{R}^+$ by an $\omega$-parametric analytic family better than order of $(\omega\ln\omega)^{-(m+1)}$. Next, we prove an estimation of the eigenvalues and characteristic values of a Sturm-Liouville operator and some properties of the solution of a certain integral equation. This allows us to deduce from [Henkin-Novikova] some positive results about the best reconstruction formula by giving an almost optimal formula of order of $\omega^{-m}$.