zbMATH — the first resource for mathematics

An \(n\)-dimensional Borg-Levinson theorem. (English) Zbl 0644.35095
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 G. Borg, Acta Math. 78, 1-96 (1946) and 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 \({\mathbb{R}}^ n.\)
Reviewer: J.Appell

35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text: DOI
[1] Agmon, S.: Spectral properties of Schr?dinger operators and scattering theory. Ann. Sc. Norm. Super Pisa. (4)2, 151-218 (1975) · Zbl 0315.47007
[2] Agmon, S., Hormander, L.: Asymptotic properties of solutions of differential equations with simple characteristics. J. Anal. Math.30, 1-38 (1976) · Zbl 0335.35013 · doi:10.1007/BF02786703
[3] Borg, G.: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta. Math.78, 1-96 (1946) · Zbl 0063.00523 · doi:10.1007/BF02421600
[4] Gelfand, I. M. Levitan, B. M.: On the determination of a differential equation from its spectral function. Izv. Akad Nauk. SSSR, Ser. Mat.15, 309-360 (1961)
[5] Levinson, N.: The inverse Sturm-Liouville problem. Mat. Tidsskr. B. 1949 25-30 (1949) · Zbl 0045.36402
[6] Lavine, R. B., Nachman, A. I.: Exceptional points in multidimensional inverse problems (in preparation)
[7] Sylvester, J., Uhlmann, G.: A uniqueness theorem for an inverse boundary value problem in electrical prospection. Commun. Pure. Appl. Math.39, 91-112 (1986) · Zbl 0611.35088 · doi:10.1002/cpa.3160390106
[8] Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math.125, 153-169 (1987) · Zbl 0625.35078 · doi:10.2307/1971291
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.