Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data

TL;DR

This paper establishes a H"older stability estimate for the multi-dimensional Borg-Levinson inverse spectral problem, even with partial and asymptotic spectral data, advancing understanding of potential recovery.

Contribution

It provides the first H"older stability estimate for the multi-dimensional Borg-Levinson theorem with partial spectral data, including asymptotic cases.

Findings

H"older stability estimate proven for partial spectral data

Stability holds even with finitely many unknown eigenvalues and derivatives

Asymptotic spectral data up to O(k^{-a}) still yields stability

Abstract

We prove a stability estimate related to the multi-dimensional Borg-Levinson theorem of determining a potential from spectral data: the Dirichlet eigenvalues and the normal derivatives of the eigenfunctions on the boundary of a bounded domain. The estimate is of H\"older type, and we allow finitely many eigenvalues and normal derivatives to be unknown. We also show that if the spectral data is known asymptotically only, up to $O(k^{-a})$ with $a>>1$, then we still have H\"older stability.