Inverse methods from scattering theory applied to a spherical plasma

A method is presented for solving the inverse problem for a set of second order ordinary differential equations given data at finite distance from the origin. The form of these equations is readily related to the radial time-independent Schroedinger equation and in turn to the linearized radial equation of a spherically or cylindrically symmetric cold plasma which has been perturbed by an oscillatory electrical potential, of fixed frequency, applied to its surface. Subject to certain open questions of stability, the inverse problem for the plasma is solved; i.e., its (heavy) ion distribution is constructed from the normal logarithmic derivative of the electric potential at the surface. This method is motivated by R. G. Newton's approach to the fixed energy inverse scattering problem for spherically symmetric potentials in the Schroedinger equation. In particular a representation is given of the regular solutions of the radial equation which involves a set of intermediary constants via which the data can be fed in, and from which both the Schroedinger potential V/sub s/(r), and the radial solutions can be generated. The derivation is elementary in nature and assumes that r2V/sub s/(r) is analytic in r/sup a/ near r = 0 with Real (a) greater than zero. Some generalizations are also given. (U.S.)