Identification of unbounded electric potentials through asymptotic boundary spectral data

TL;DR

This paper proves that the electric potential within a bounded domain can be uniquely identified from the asymptotic behavior of eigenvalues and boundary measurements of eigenfunctions, even for unbounded potentials.

Contribution

It establishes a uniqueness result for determining unbounded electric potentials from spectral boundary data in higher dimensions.

Findings

Unique determination of electric potential from spectral data.

Extension to unbounded potentials in higher dimensions.

Asymptotic boundary spectral data suffices for identification.

Abstract

We prove that the real-valued electric potential $q \in L^{\max(2,3 n / 5)}(\Omega)$ of the Dirichlet Laplacian $-\Delta +q$ acting in a bounded domain $\Omega \subset \mathbb{R}^n$, $n \ge 3$, is uniquely determined by the asymptotics of the eigenpairs formed by the eigenvalues and the boundary observation of the normal derivative of the eigenfunctions.