Some remarks on the multi-dimensional Borg-Levinson theorem. (English) Zbl 0708.35094

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1989, Exp. No. 17, 6 p. (1989).
As was shown by A. Nachman, J. Sylvester, and G. Uhlmann [Commun. Math. Phys. 115, No.4, 595-605 (1988; Zbl 0644.35095)], the potential \(q\in C^{\infty}({\bar \Omega})\) in the problem \[ -\Delta u+q(x)u=\lambda u(\Omega),\quad u=0\quad (\partial \Omega) \] is uniquely determined by the spectrum \(\lambda_ 1<\lambda_ 2\leq...\leq \lambda_ n\leq..\). of \(-\Delta +q\), and the “Dirichlet-Neumann map”. Here the author proves that the same is true for all but finitely many \(\lambda_ j\). Thus, the higher-dimensional inverse problem is more “rigid” than the scalar one [cf. G. Borg, Acta Math. 78, 1-96 (1946)].
Reviewer: J.Appell


35R30 Inverse problems for PDEs
35P05 General topics in linear spectral theory for PDEs
35J10 Schrödinger operator, Schrödinger equation


Zbl 0644.35095
Full Text: Numdam EuDML