Multidimensional Borg--Levinson theorems for unbounded potentials

TL;DR

This paper establishes the uniqueness of determining unbounded potentials in a Schrödinger operator from spectral data, extending classical inverse spectral results to cases with incomplete data and unbounded potentials.

Contribution

It proves multidimensional Borg--Levinson theorems for unbounded potentials, including cases with incomplete spectral data, for the first time in higher dimensions.

Findings

Dirichlet eigenvalues and boundary data determine the potential for $q \,\in\, L^{n/2}$.

Uniqueness holds even with incomplete spectral data for certain $L^p$ spaces.

Results extend inverse spectral theory to unbounded potentials in multiple dimensions.

Abstract

We prove that the Dirichlet eigenvalues and Neumann boundary data of the corresponding eigenfunctions of the operator $-\Delta + q$, determine the potential $q$, when $q \in L^{n/2}(\Omega,\mathbb{R})$ and $n \geq 3$. We also consider the case of incomplete spectral data, in the sense that the above spectral data is unknown for some finite number of eigenvalues. In this case we prove that the potential $q$ is uniquely determined for $q \in L^p(\Omega,\mathbb{R})$ with $p=n/2$, for $n\geq4$ and $p>n/2$, for $n=3$.