Principle of Maximum Entropy and Inverse Scattering Problems

Using the principle of maximum entropy, a procedure is outlined to study some aspects of the inverse scattering problem. As an application we study a) the quantum mechanical inverse scattering problem using the Born Approximation, b) the electromagnetic inverse problem, and c) the solutions to the Marchenko equation of inverse scattering. INTRODUCTION Following the pioneering work by Jaynes [1], concerning the information theoretic approach to statistical mechanics, there have been several novel applications of the principle of maximum entropy in a number of areas such as image processing, and geophysical data analysis [2]. Of particular importance to the present work is a paper by Jaynes on time series analysis using the principle of maximum entropy in which Jaynes [3] derived Burg's [4] spectral method. The main goal of this paper is to demonstrate the applicability of the maximum entropy method for analyzing the generalized inverse problem. In Section 2 we develop the formulation for tackling generalized inverse problems suitable for the case when the available information is either incomplete or noisy. The efficacy of the formulation of Section 2 is demonstrated in Section 3 by studying its application to the inverse problem in quantum mechanical scattering theory. We present two more examples in Sections 4 and 5 where we present a solution to the inverse problem associated with the Marchenko integral equation of scattering theory and the electromagnetic inverse problem. Finally, in Section 6 we make some concluding remarks.