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An inverse spectral problem in three dimensions. (English) Zbl 0554.35125
Inverse problems, Proc. Symp. Appl. Math., New York 1983, SIAM-AMS Proc. 14, 81-90 (1984).
[For the entire collection see Zbl 0534.00010.]
Author’s abstract: This paper summarizes a generalization of the Gel’fand-Levitan method for the solution of the inverse spectral problem of the Schrödinger equation in \({\mathbb{R}}^ 3\) that had been given by the author in a series of papers in 1980 to 1982. Whereas in \({\mathbb{R}}\) and in \({\mathbb{R}}_+\) a ”regular” solution may be defined by a boundary condition at a point, in \({\mathbb{R}}^ n\), \(n>1\), this cannot be done, and the very definition of the solution-family whose spectral weights form the starting point of the Gel’fand-Levitan inversion is part of the problem. The author provides such a definition by means of a Riemann- Hilbert problem whose solution, a generalized Jost function, leads from the scattering solution of the Schrödinger equation to the ”regular” solution, which is an entire function of the wave number. The resulting generalization of the Gel’fand-Levitan equation is a singular integral equation with distributional solutions.
MSC:
35R30 Inverse problems for PDEs
35J10 Schrödinger operator, Schrödinger equation
35P25 Scattering theory for PDEs