Extensions of Variational Methods, III

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Abstract
The potential V(r) is uniquely constructed from the phase shift function ηl(k) in any fixed value l of angular momentum under a certain condition on V(r), if the system has no bound state. The usual variational method is extended so as to give a variation principle for V(r) in that case. We will consider an integral equation in the following form where A(k) and B(k, r) are determined by a trial function which is specified by a given ηl(k). By means of this equation it is shown that V(r) can be calculated for any r(≧0) accurately within errors of the order of Δ2, while the errors involved in the trial function are of the order Δ. When the system has bound states the situation is more complicated, but the phase shift ηl(k) uniquely determines the potential for the system as well as the binding energies of all bound states, when we restrict ourselves to a potential of short range which is subjected to the condition, where -κ is the logarithmic derivative for the asymptotic wave function of the lowest level. The arguments given in the case of no bound state remain still valid for such a short range potential. Some generalizations of our method are proposed to include the tensor potential as well as the central one, which will enable us to make use of known phase shifts in the two-nucleon problem.

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