An \(n\)-dimensional Borg-Levinson theorem
type: Item
instance of: scholarly article
title: An \(n\)-dimensional Borg-Levinson theorem
zbMATH Open document ID: 0644.35095
DOI: 10.1007/BF01224129
author: Adrian Nachman
author: John Sylvester
author: Gunther Uhlmann
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: Let \(y=y(x;\lambda,q)\) denote the solution of the Sturm-Liouville problem \(-y''+q(x)y=\lambda y\) over \(\Omega =(0,1)\), subject to initial conditions \(y(0,\lambda)=0\), \(y'(0,\lambda)=1\). It is well known that the Dirichlet eigenvalues \(\mu_ n=\mu_ n(q)\) (i.e. the \(\mu\) 's satisfying \(y(1;\mu,q)=0)\) and the ``normalizing constants'' \(k_ n=k_ n(q)=y'(1;\mu_ n(q),q)\) together determine the potential q uniquely on [0,1] [see \textit{G. Borg}, Acta Math. 78, 1-96 (1946) and \textit{N. Levinson}, Mat. Tidsskr. B 1949, 25-30 (1949; Zbl 0045.364)]. In the present paper, this is generalized to the higher dimensional case as follows. Given a bounded smooth domain \(\Omega\) in \({\mathbb{R}}^ n \)and \(q\in C^{\infty}({\bar \Omega})\), let \(\mu_ n=\mu_ n(q)\) denote the eigenvalues and \(\phi_ n=\phi_ n(x;q)\) (some) eigenfunctions of the Dirichlet problem \[ (*)\quad -\Delta u+qu=\lambda u\quad (in\quad \Omega),\quad u\equiv 0\quad (\text{on } \partial \Omega). \] Then the sequences \(\mu_ n=\mu_ n(q)\) and \(k_ n=k_ n(q)=(\partial /\partial \nu)\phi_ n(\cdot;q)|_{\partial \Omega}\) determine the potential q uniquely on \({\bar \Omega}\). The proof builds on a careful analysis of the Green's function of the ``iterates'' of (*), as well as norm estimates for ``scattering'' solutions of (*) on \({\mathbb{R}}^ n.\)
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 4050176
zbMATH Keywords: Sturm-Liouville problem
zbMATH Keywords: Dirichlet eigenvalues
zbMATH Keywords: normalizing constants
zbMATH Keywords: Dirichlet problem
reviewed by: Jürgen Appell
MaRDI profile type: Publication
cites work: Q4076370
cites work: Asymptotic properties of solutions of differential equations with simple characteristics
cites work: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte
cites work: Q5810726
cites work: A uniqueness theorem for an inverse boundary value problem in electrical prospection
cites work: A global uniqueness theorem for an inverse boundary value problem
full work available at URL: https://doi.org/10.1007/bf01224129
OpenAlex ID: W2058356772
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description: scientific article; zbMATH DE number 4050176