ar X iv : m at h - ph / 9 91 10 32 v 1 2 5 N ov 1 99 9 PROPERTY C FOR ODE AND APPLICATIONS TO INVERSE PROBLEMS

An overview of the author's results is given. Property C stands for completeness of the set of products of solutions to homogeneous linear Sturm-Liouville equations. The inverse problems discussed include the classical ones (inverse scattering on a half-line, on the full line, inverse spectral problem), inverse scattering problems with incomplete data, for example, inverse scattering on the full line when the reflection coefficient is known but no information about bound states and norming constants is available, but it is a priori known that the potential vanishes for x < 0, or inverse scattering on a half-line when the phase shift of the s-wave is known for all energies, no bound states and norming constants are known, but the potential is a priori known to be compactly supported. If the potential is compactly supported, then it can be uniquely recovered from the knowledge of the Jost function f (k) only, or from f ′ (0, k), for all k ∈ ∆, where ∆ is an arbitrary subset of (0, ∞) of positive Lebesgue measure. Inverse scattering problem for an inhomogeneous Schrödinger equation is studied. Inverse scattering problem with fixed-energy phase shifts as the data is studied. Some inverse problems for parabolic and hyperbolic equations are investigated. A detailed analysis of the invertibility of all the steps in the inversion procedures for solving the inverse scattering and spectral problems is presented. An analysis of the Newton-Sabatier procedure for inversion of fixed-energy phase shifts is given. Inverse problems with mixed data are investigated. Representation formula for the I-function is given and properties of this function are studied. Algorithms for finding the scattering data from the I-function, the I-function from the scattering data and the potential from the I-function are given. A characterization of the Weyl solution and a formula for this solution in terms of Green's function are obtained.