Nonlocal potentials and complex angular momentum theory

Abstract

The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T(lambda,k) associated with physically relevant classes N_{w,alpha}^gamma of nonlocal potentials in corresponding domains D_{gamma,alpha}^delta of the space C^2 of the complex angular momentum lambda and of the complex momentum k (namely, the square root of the energy). The general expression of T as a quotient Theta(lambda,k)/sigma(lambda,k) of two holomorphic functions in D_{gamma,alpha}^delta is obtained by using the Fredholm-Smithies theory for complex k, at first for lambda=l integer, and in a second step for lambda complex (Real(lambda)>-1/2). Finally, we justify the "Watson resummation" of the partial wave amplitudes in an angular sector of the lambda-plane in terms of the various components of the polar manifold of T with equation sigma(lambda,k)=0. While integrating the basic Regge notion of interpolation of resonances in the upper half-plane of lambda, this unified representation of the singularities of T also provides an attractive possible description of antiresonances in the lower half-plane of lambda. Such a possibility, which is forbidden in the usual theory of local potentials, represents an enriching alternative to the standard Breit-Wigner hard-sphere picture of antiresonances.