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

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.