Stable determination of unbounded potential by asymptotic boundary spectral data

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 $q\in L^{\max (2,3d/5)}(\Omega )$. 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 Hölder stably retrieved through knowledge of the asymptotics of the eigenvalues