Identification of piecewise-constant potentials by fixed-energy phase shifts

The identification of a spherically symmetric potential by its phase shifts is an important physical problem. Recent theoretical results assure that s uch a potential is uniquely defined by a sufficiently large subset of its p hase shifts at any one fixed energy level. However, two different potential s can produce almost identical phase shifts. To resolve this difficulty we suggest the use of phase shifts corresponding to several energy levels. The identification is done by a nonlinear minimization of the appropriate obje ctive function. It is based on a combination of probabilistic global and de terministic local minimization methods. The Multilevel Single-Linkage Metho d (MSLM) is used for the global minimization. A specially designed Local Mi nimization Method (LMM) with a Reduction Procedure is used for the local se arches. Numerical results show the effectiveness of this procedure for pote ntials composed of a small number of spherical layers.