Appl Math Optim 44:49–65 (2001)
2001 Springer-Verlag New York Inc.
Identiﬁcation of Piecewise-Constant Potentials
by Fixed-Energy Phase Shifts
Department of Mathematics, University of Oklahoma,
Norman, OK 73019, USA
Abstract. The 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. 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 objective function. It is based on a combination of
probabilistic global and deterministic local minimization methods. The Multilevel
Single-Linkage Method (MSLM) is used for the global minimization. A specially
designed Local Minimization Method (LMM) with a Reduction Procedure is used
for the local searches. Numerical results show the effectiveness of this procedure
for potentials composed of a small number of spherical layers.
Key Words. Inverse scattering, Phase shifts, Global minimization.
AMS Classiﬁcation. Primary 35R30, 65K10, Secondary 86A22.
Let q(x), x ∈ R
, be areal-valuedpotential with compactsupport. Let R > 0be such that
q(x) = 0 for
> R. We also assume that q ∈ L
≤ R, x ∈ R
be the unit sphere, and α ∈ S
. For a given energy k > 0 the scattering solution
ψ(x,α)is deﬁned as the solution of
ψ + k
(1 − q(x))ψ = 0 (1.1)