Holomorphic Extensions Associated with Fourier–Legendre Series and the Inverse Scattering Problem

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