Multidimensional inverse scattering problems

Abstract

Summary form only given, as follows. An overview of the author's results is given. The inverse problems for obstacle, geophysical and potential scattering are considered. The basic method for proving uniqueness theorems in one- and multi-dimensional inverse problems is discussed and illustrated by numerous examples. The method is based on property C for pairs of differential operators. Property C stands for completeness of the sets of products of solutions to homogeneous differential equations. To prove a uniqueness theorem in the inverse scattering problem one assumes that there are two operators which generate the same scattering data. This assumption allows one to derive an orthogonality relation from which, via property C, the uniqueness theorem follows. New results are discussed. These include property C for ordinary differential equations, inversion of I-function (impedance function), inversion of incomplete scattering data (for example, the phase shift of s-wave without the knowledge of bound states and norming constants but assuming a priori that the potential has compact support, etc). Analytical solution of the ground-penetrating radar problem is outlined. Open problems are formulated.