Inverse problems for Sturm--Liouville operators with potentials from Sobolev spaces. Uniform stability

TL;DR

This paper establishes uniform stability estimates for inverse Sturm--Liouville problems with potentials in Sobolev spaces, including classical cases, by analyzing spectral data and associated mappings.

Contribution

It introduces a new framework for analyzing inverse problems with Sobolev space potentials, providing uniform stability results for spectral data reconstruction.

Findings

Uniform stability estimates for potential recovery from spectral data.

Extension of results to classical case q in L2.

Construction of specialized Hilbert spaces for spectral data.

Abstract

The paper deals with two inverse problems for Sturm--Liouville operator $Ly=-y" +q(x)y$ on the finite interval $[0,\pi]$. The first one is the problem of recovering of a potential by two spectra. We associate with this problem the map $F:\, W^\theta_2\to l_B^\theta,\ F(\sigma) =\{s_k\}_1^\infty$, where $W^\theta_2 = W^\theta_2[0,\pi]$ are Sobolev spaces with $\theta\geqslant 0$, $\sigma=\int q$ is a primitive of the potential $q$ and $l_B^\theta$ are special Hilbert spaces which we construct to place in the regularized spectral data $\bold s = \{s_k\}_1^\infty$. The properties of the map $F$ are studied in details. The main result is the theorem on uniform stability. It gives uniform estimates from above and below of the norm of the difference $\|\sigma -\sigma_1\|_\theta$ by the norm of the difference of the regularized spectral data $\|\bold s -\bold s_1\|_\theta$ where the last norm is taken in $l_B^\theta$. A similar result is obtained for the second inverse problem when the potential is recovered by the spectral function of the operator $L$ generated by Dirichlet boundary conditions. The results are new for classical case $q\in L_2$ which corresponds to the value $\theta =1$.