Stable determination of unbounded potential by asymptotic boundary spectral data

TL;DR

This paper proves that the electric potential in a bounded domain can be stably reconstructed from the asymptotic behavior of eigenvalues of the Dirichlet Laplacian, given certain boundary measurements, even for unbounded potentials.

Contribution

It establishes a Hölder stability result for the inverse spectral problem of determining unbounded potentials from asymptotic eigenvalue data.

Findings

Potential can be stably recovered from eigenvalue asymptotics.

Hölder stability holds under boundary measurement conditions.

Results apply to unbounded potentials in specified Lebesgue spaces.

Abstract

We consider the Dirichlet Laplacian $A_q=-\Delta+q$ in a bounded domain $\Omega \subset \mathbb{R}^d$, $d \ge 3$, with real-valued perturbation $q \in L^{\max(2 , 3 d / 5)}(\Omega)$. We examine the stability issue in the inverse problem of determining the electric potential $q$ from the asymptotic behavior of the eigenvalues of $A_q$. Assuming that the boundary measurement of the normal derivative of the eigenfunctions is a square summable sequence in $L^2(\partial \Omega)$, we prove that $q$ can be H\"older stably retrieved through knowledge of the asymptotics of the eigenvalues