Mathematical Sciences: Inverse Scattering Problems

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Abstract Ralston/Eskin 9622310 In this proposal we present several inverse coefficient problems which we plan to attack using the methods introduced in our paper (Commun. Math. Phys.173 (1995)). In most cases we wish to recover the unknown coefficients from scattering data (or equivalently the Dirichlet to Neumann map) at a fixed energy. The coefficients are usually assumed to approach limiting values exponentially, but are not necessarily constant outside compact sets. The equations we plan to study are the Schroedinger equation with an external gauge potential, the Laplace-Beltrami equation for a metric which tends exponentially to the Euclidean metric, Maxwell's equations, and the Schroedinger equation with magnetic and electric potentials in two space dimensions. Inverse scattering problems arise when one wishes to recover physical properties of an object from data obtained by remote observations. Interpreted broadly these problems include methods of nondestructive testing and medical imaging. The problems we propose to study include recovering the electrical properties of a medium from signals sent through it, and the reconstruction of the forces acting on a quantum particle from the observation of the deflection of its trajectories.