  An identification of a spherically symmetric potential by its phase shifts is
an important physical problem. Recent theoretical results assure that such a
potential is uniquely defined by a sufficiently large subset of its phase
shifts at any one fixed energy level. However, two different potentials can
produce almost identical phase shifts. That is, the inverse problem of the
identification of a potential from its phase shifts at one energy level $k^2$
is ill-posed, and the reconstruction is unstable. In this paper we introduce a
quantitative measure $D(k)$ of this instability. The diameters of minimizing
sets
  $D(k)$ are used to study the change in the stability with the change of $k$,
and the influence of noise on the identification. They are also used in the
stopping criterion for the nonlinear minimization method IRRS (Iterative Random
Reduced Search). IRRS combines probabilistic global and deterministic local
search methods and it is used for the numerical recovery of the potential by
the set of its phase shifts. The results of the identification for noiseless as
well as noise corrupted data are presented.
