Recovering of a potential of Sturm-Liouville operator from a finite sets of eigenvalues and norming constants

TL;DR

This paper establishes a quantitative approximation method for reconstructing a Sturm-Liouville potential from finite spectral data, providing explicit error bounds in Sobolev norms based on the number of eigenvalues and norming constants used.

Contribution

It introduces a finite-data approximation scheme for Sturm-Liouville potentials with explicit error estimates in Sobolev spaces, extending previous unique recovery results.

Findings

Error estimate $ orm{q - q_N}_ au o 0$ as $N o fty$

Approximation accuracy depends on the number of spectral data points and the smoothness of the potential

Provides bounds uniform over potentials with bounded Sobolev norm

Abstract

It is well known that a potential $q$ of the Sturm-Liouville operator $Ly= -y" +q(x)y$ on the finite interval $[0, \pi]$ can be uniquely recovered by the spectrum $\{\lambda_k\}_1^\infty$ and norming constants $\{\alpha_k\}_1^\infty$ of this operator with Dirichlet boundary conditions. Given potential $q$ belonging to Sobolev space $W^\theta_2[0, \pi]$ with $\theta > -1$ we associate its $2N$-approximation $q_N$ constructed by the final sets $\{\lambda_k\}_1^N$ and $\{\alpha_k\}_1^N$. The main result claims that for $-1\leqslant\tau <\theta$ the estimate $\|q -q_N\|_\tau \leqslant CN^{\theta-\tau}$ holds, where $\|\cdot\|_\tau$ is the norm in $W^\tau_2$ and the constant $C$ depends on $R$ but does not depend on $q$ if $\|q\|_\theta \leqslant R$.