The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T(λ, k) associated with physically relevant classes of nonlocal potentials in corresponding domains of the space of the complex angular momentum λ and of the complex momentum k (namely, the square root of the energy). The general expression of T as a quotient Θ(λ, k)/σ(λ, k) of two holomorphic functions in is obtained by using the Fredholm–Smithies theory for complex k, at first for λ = ℓ integer, and in a second step for λ complex (Re λ > −1/2). Finally, we justify the “Watson resummation” of the partial wave amplitudes in an angular sector of the λ-plane in terms of the various components of the polar manifold of T with equation σ(λ, k) = 0. While integrating the basic Regge notion of interpolation of resonances in the upper half-plane of λ, this unified representation of the singularities of T also provides an attractive possible description of echoes in the lower half-plane of λ. 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.
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