Some remarks on the multi-dimensional Borg-Levinson theorem
type: Item
instance of: scholarly article
title: Some remarks on the multi-dimensional Borg-Levinson theorem
zbMATH Open document ID: 0753.35121
author: H. Isozaki
published in: Journal of Mathematics of Kyoto University
publication date: 1992-09-27T00:00:00Z
review text: In a recent paper [Commun. Math. Phys. 115, No. 4, 595-605 (1988; Zbl 0644.35095)], \textit{A. I. Nachman}, \textit{J. Sylvester} and \textit{G. Uhlmann} have obtained an extension of the classical Borg-Levinson theorem to the inverse problem of recovering the potential \(q\) in \[ -\Delta u+q(x)u=\lambda u \qquad (x\in\Omega),\tag{*} \] subject to Dirichlet boundary conditions. Loosely speaking, they showed that the potential is determined by the eigenvalues and the normal derivatives of the eigenfunctions. In the present paper, the author shows that, for smooth potentials, the knowledge of all but finitely many eigenvalues and eigenfunctions suffices. This shows that the higher-dimensional inverse problem (*) is ``more rigid'' than its scalar analogue \[ -y''+q(x)y=\lambda y\qquad (0\leq x\leq 1). \]
reviewed by: Jürgen Appell
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34A55
zbMATH DE Number: 59740
zbMATH Keywords: uniqueness in inverse eigenvalue problems
zbMATH Keywords: smooth potentials
zbMATH Keywords: higher- dimensional inverse problem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519727
OpenAlex ID: W1597597747
DOI: 10.1215/KJM/1250519727
description: scientific article; zbMATH DE number 59740