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2000-2001
MATHEMATICS COLLOQUIUM SERIES
Property C for PDE and ODE and Applications
Abstract: A general method, introduced by the author for a study of a wide variety of inverse problems, is described. The method is based on property C for pairs of operators, that is, completeness of the set of products of the elements of the null spaces of these operators. This method was applied to the inverse scattering problems with fixed-energy data and to many other inverse problems. Property C for ODE is introduced and some of its applications are presented. A simple and short proof is given for some classical results, for example, uniqueness of the solution to inverse scattering problem on the half-axis, and of some new results, for example, uniqueness of the solution to inverse scattering problem on the half-axis when the scattering data are incomplete and the potential is compactly supported. Some inverse problems for parabolic and hyperbolic problems are considered.