Nonlocal potentials and complex angular momentum theory

The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T(lambda, k) associated with physically relevant classes N-omega(epsilon),alpha(gamma) a 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 (Re 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 echoes 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 echoes.