Holomorphic extensions associated with Fourier-Legendre series and the inverse scattering problem

In this paper, we consider the inverse scattering problem and, in particular, the problem of reconstructing the spectral density associated with the Yukawian potentials from the sequence of the partial-waves f_l of the Fourier-Legendre expansion of the scattering amplitude. We prove that if the partial-waves f_l satisfy a suitable Hausdorff-type condition, then they can be uniquely interpolated by a complex function f(lambda), analytic in a half-plane. Assuming also the Martin condition to hold, we can prove that the Fourier-Legendre expansion of the scattering amplitude converges uniformly to a function f (theta) in C (theta being the complexified scattering angle), which is analytic in a strip contained in the theta-plane. This result is obtained mainly through geometrical methods by replacing the analysis on the complex cos theta-plane with the analysis on a suitable complex hyperboloid. The double analytic symmetry of the scattering amplitude is therefore made manifest by its analyticity properties in the lambda-and theta-planes. The function f (theta) is shown to have a holomorphic extension to a cut-domain, and from the discontinuity across the cuts we can iteratively reconstruct the spectral density sigma(µ) associated with the class of Yukawian potentials. A reconstruction algorithm which makes use of Pollaczeck and Laguerre polynomials is finally given.