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Some remarks on the multi-dimensional Borg-Levinson theorem. (English) Zbl 0753.35121
In a recent paper [Commun. Math. Phys. 115, No. 4, 595-605 (1988; Zbl 0644.35095)], A. I. Nachman, J. Sylvester and 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 \[ -y''+q(x)y=\lambda y\qquad (0\leq x\leq 1). \]

35R30 Inverse problems for PDEs
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34A55 Inverse problems involving ordinary differential equations
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