Unitarity constraints with the Kapur-Peierls formalism

Abstract

It is well known that the Kapur-Peierls formalism, as currently used for the analysis of resonance reactions, fails to provide a manifestly unitary collision matrix expansion, in contrast to the $R$-matrix theory. This drawback, and the presence of a number of parameters generally larger than in the $R$-matrix theory, inhibits the general use of the Kapur-Peierls formalism, except for special cases in which it was possible to establish explicit relations to the $R$-matrix formalism. It is shown here that the basic properties of the Kapur-Peierls eigenfunctions provide relations among the parameters, which express in general the condition of unitarity. These relations, obtained without need of imposing limits to the number of resonances and of reaction channels, and retaining the general nature of the parameters of being energy dependent, establish constraints among the Kapur-Peierls parameters. As a result, the number of real independent parameters for the Kapur-Peierls formalism is reduced to be the same as in the $R$-matrix theory. From the same relations is derived also a transformation from the Kapur-Peierls parameters to the $R$-matrix parameters, in an equally general form. The basic feature of unitarity for the Kapur-Peierls parameters is retained if in applications only limited numbers of resonances and channels are introduced. Analytical procedures based on an iteration-perturbation scheme are presented so as to provide for the explicit use of the unitarity constraints in numerical applications and data analysis. The same procedures provide $R$-matrix parameters unitarily related to the Kapur-Peierls parameters, and representing the same collision matrix.NUCLEAR REACTIONS Kapur-Peierls formalism, unitarity constraints, iteration procedures in numerical applications, data analysis.