Property C for ODE and applications to inverse problems

Abstract

An overview of the author's results is given. Property C stands for completeness of the set of products of solutions to homoge- neous 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 incom- plete data, for example, inverse scattering on the full line when the reection coecient 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 (0;k), for all k 2 , where is an arbitrary subset of (0;1) of positive Lebesgue measure. Inverse scattering problem for an inhomogeneous Schrodinger equa- tion is studied. Inverse scattering problem with xed-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 xed- 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 nding 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 so- lution in terms of Green's function are obtained.