Global method for all S-matrix poles identification new classes of poles and resonant states

Abstract

A global method to identify all the S-matrix poles k = k l ( g) in the k-plane for a central potential gV( r) ( g ϵ C ) is presented. The method involves construction of the Riemann surface R g ( l) over the g-plane, on which the function k = k l ( g) is single valued and analytic. It implies the division of the Riemann surface R g ( l) into Σ n ( l) sheets and the construction of the sheet images Σ′ n ( l) in the k-plane. The method is first applied to potentials having Jost entire functions in both variables g and k. In this case the general properties of the function k = k l ( g) and of its Riemann surface R g ( l) are obtained. The method is then extended to other classes of potentials. The construction of the Riemann surface R g ( l) as well as the construction of the images of its sheets in the k-plane for some particular potentials is given in detail. The method is allowed to identify new classes of resonant state poles. These poles remain in a neighborhood of some special points called “stable-points” as the strength of the potential increases to infinity; i.e., these resonant state poles do not become bound or virtual state poles. Moreover, the wave functions of the resonant states corresponding to the new-class poles situated in the neighborhood of the stable-points are almost completely localized outside the potential well. The method provides a quantum number n, characterizing the bound and resonant state S-matrix poles. This quantum number has a topological significance: it is the label of the sheets of the Riemann surface R g ( l) . The existence of new classes of resonant state poles seems to be a general property of the potentials with barriers used in nuclear scattering theory.