Multidimensional Borg-Levinson inverse spectral theory

TL;DR

This paper proves that in multidimensional bounded domains, the Dirichlet eigenvalues and Neumann boundary data uniquely determine the potential in the inverse spectral problem, even with incomplete or asymptotic data, and applies this to parabolic inverse problems.

Contribution

It extends multidimensional Borg-Levinson inverse spectral theory to cases with incomplete or asymptotic spectral data and applies it to solve inverse coefficient problems for parabolic equations.

Findings

Unique determination of potential from spectral data in multidimensions.

Recovery of potential with incomplete spectral information under summability conditions.

Application of spectral theory to inverse parabolic coefficient problems.

Abstract

This text deals with multidimensional Borg-Levinson inverse theory. Its main purpose is to establish that the Dirichlet eigenvalues and Neumann boundary data of the Dirichlet Laplacian acting in a bounded domain of dimension 2 or greater, uniquely determine the real-valued bounded potential. We first address the case of incomplete spectral data, where finitely many boundary spectral eigen-pairs remain unknown. Under suitable summability condition on the Neumann data, we also consider the case where only the asymptotic behavior of the eigenvalues is known. Finally, we use the multidimensional Borg-Levinson theory for solving parabolic inverse coefficient problems.