Article

Quantum mechanical inverse scattering problem at fixed energy: a constructive method

11/2011; DOI: 10.4310/MAA.2011.v18.n1.a6
Source: arXiv

ABSTRACT The inverse scattering problem of the three-dimensional Schroedinger equation
is considered at fixed scattering energy with spherically symmetric potentials.
The phase shifts determine the potential therefore a constructive scheme for
recovering the scattering potential from a finite set of phase shifts at a
fixed energy is of interest. Such a scheme is suggested by Cox and Thompson [3]
and their method is revisited here. Also some new results are added arising
from investigation of asymptotics of potentials and concerning statistics of
colliding particles. A condition is given [2] for the construction of
potentials belonging to the class L_1,1 which are the physically meaningful
ones. An uniqueness theorem is obtained [2] in the special case of one given
phase shift by applying the previous condition. It is shown that if only one
phase shift is specified for the inversion procedure the unique potential
obtained by the Cox-Thompson scheme yields the one specified phase shift while
the others are small in a certain sense. The case of two given phase shifts is
also discussed by numerical treatment and synthetic examples are given to
illustrate the results. Besides the new results this contribution provides a
systematic treatment of the CT method.

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