Analytic Properties of a Class of Potentials and the Corresponding Jost Functions

Abstract

Analytical properties of the potential obtained by Newton's method in the inverse scattering problem at fixed energy are thoroughly investigated. It is found that r V(r) is analytic in the neighborhood of r = 0 and can be continued in the r complex plane as a meromorphic function, the poles of which are of order at least equal to 2. An explicit formula is given to yield the Jost functions for any complex value of the angular momentum ν. They appear to be meromorphic functions with poles located at the negative integers on the real axis. The zeros of f₂(ν), for large |ν| and for Re ν and Im ν > 0, are located on a curve which is the boundary of a domain previously shown to contain no Regge pole. The interpolated scattering amplitude is unitary. It behaves for ν → −i ∞ as e^π|ν|. An important result of this paper is that the class of potentials obtained by Newton's method is much more restricted than one might think. This led the author to look for a more general approach to the inverse scattering problem, which is the subject of a forthcoming publication. In order to illustrate the method of this paper, a detailed study of an example previously introduced by R. G. Newton is given. In the last Appendix, a very remarkable property of the potentials involved in this example is given: the scattering amplitude corresponding to these potentials can be given an exact closed form. Since the corresponding potentials are strongly energy‐dependent, it is very likely that this is only a mathematical curiosity.