Asymptotic Properties of the Potentials in the Inverse-Scattering Problem at Fixed Energy

Abstract

The problem of finding a nonrelativistic central potential from a knowledge of all the phase shifts at one energy had been previously shown by Newton to reduce to the inversion of a given infinite matrix M. In the framework of Newton's theory, the solution is not unique but depends on one parameter. In the present work, the inverse matrix is explicitly given, together with the vectors annihilated by M. These enable one to construct all the solutions of the problem. The asymptotic behavior of the equivalent potentials is exhibited, and it is shown that one (and only one) of them decreases asymtotically faster than r^−2+ε, provided that the phase shifts decrease asymptotically faster than l^−(3+ε′) (for arbitrarily small ε, ε′). All the other equivalent potentials have an oscillating tail damped by a factor r^−3/2. The ``transparent potentials,'' which give all phase shifts equal to zero at one energy, are also studied. In subsequent publications, analytic continuation of the potentials in the r plane and of the Jost function in the angular momentum complex plane is studied.