Mathematics of the N/D Method

Abstract

A complete mathematical account is given of the N/D method, subject to certain restrictions on the given functions L and ρ (left‐hand cut contribution and phase‐space factor). The restrictions are the weakest possible if the phase shift is to be Hölder continuous. Any asymptotic behavior ρ ∼ x^β, 0 < β < 2, is allowed. No analyticity is assumed for L. Exhaustive existence and uniqueness theorems are given: Any allowed function F (= e^iδ sin δ/ρ) possesses an N/D decomposition, and the integral equation for N acts in a space of L₂ functions; this equation satisfies the Fredholm alternative theorems [though its kernel has a continuous spectrum if δ(∞)/π is not integral]; any L₂ solution of the equation yields an allowed function F; all allowed solutions may be obtained by varying the CDD parameters (which enter linearly); each solution has a uniqueness index κ; there will usually be a κ parameter infinity of solutions with each κ ≥ 0, and none with κ < 0, but precise conditions on L and ρ are given in order that there will be a negative κ solution.