QUANTUM INVERSE SCATTERING PROBLEM AS A CAUCHY-PROBLEM

An approach to the inverse problem of quantum scattering at fixed angular momentum 1l using new nonlinear equations is proposed. In this approach, energy levels, normalization constants, and the Jost function of the problem on the interval with a variable left bounary [r, ∞) are considered. These functions as functions of r numbered by energy E as an index (discrete or continuous) satisfy the infinite system of ordinary first-order differential equations. The scattering data serve as initial conditions for this system, and the inverse scattering problem is reduced to the Cauchy problem. As the functions considered in our treatment are slowly varying functions of r, the equations presented here are convenient for practical calculations. Some numerical examples show that the problem of reconstruction of the potential can be solved with high accuracy even with the simplest algorithms. © 1991.