Some remarks on the multi-dimensional Borg-Levinson theorem
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 \[ -y''+q(x)y=\lambda y\qquad (0\leq x\leq 1). \]
uniqueness in inverse eigenvalue problems
scientific article; zbMATH DE number 59740