Abstract
The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T(λ, k) associated with physically relevant classes \({{\mathcal{N}}^\gamma_w(\varepsilon),_{\alpha}}\) of nonlocal potentials in corresponding domains \({D_{\gamma,\alpha}^{(\delta)}}\) of the space \({\mathbb{C}^2}\) 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 \({D_{\gamma,\alpha}^{(\delta)} }\) 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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Communicated by Klaus Fredenhagen.
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Bros, J., De Micheli, E. & Viano, G.A. Nonlocal Potentials and Complex Angular Momentum Theory. Ann. Henri Poincaré 11, 659–764 (2010). https://doi.org/10.1007/s00023-010-0041-8
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DOI: https://doi.org/10.1007/s00023-010-0041-8