Analiticity and Padé method in the bound-state problem

Abstract

Recently the method of Pad~ approximants has proved to yield satisfactory resul~ in strong-interaction physics (1-4). When it is applied to the partial-wave projection of the T or S matrix Born series, in the framework of the usual Lagrangian approach or the theory of germs, the uni tar i ty condition turns out to be automatically satisfied (for diagonal approximants). The analytic properties in the square of c.m. energy are globally correct; however the Pad6 denominator can exhibit some spurious singularities, that prevent us from obtaining actual quanti tat ive information about bound states (5). The purpose of this note is to improve the content of the Padg method, concerning bound states, by giving a prescription to treat the analiticity troubles. We focus our at tention on potential theory where the validity of any approximation method can be tested with some care. & large class of regular potentials has a partialwave Born expansion exhibiting the same analytic properties as field-theory models; so the same features are present in the search of bound states through Pad6 approximants. Potential theory gives some further advantages: the convergence of the Pad6 approximants to the exact amplitude and of their normalized denominators t o the ;lost function has been proved (e-s). This suggests that the spurious singularities mentioned above should disappear by increasing the order of approximation; an explicit test of this cancellation has been made for the exponential potential (g). However, from a practical point of view it may be very laborious and difficult to calculate higher Born terms. For these reasons, in order to have an amenable method to calculate bound states, we stop with the first Pad6 approximant and improve it by a suitable recipe based on a standard technique for the analytic functions.