Theory of Regge poles for 1/r2 potentials. I

In the following series of papers we give a detailed study of the theory of Regge poles for 1/r2 potentials with the behavior r2 V (r) = −V0 at r = 0 and r2 V (r) = −V2 at r = ∞. We give a complete description of the distribution of the Regge poles in the λ plane, which is cut from −V01/2 to V01/2 and study the behavior of the pole trajectories. We find that the high energy limit of the Regge poles is controlled by the parameter p = (λ2−V0)1/2 while at low energies the relevant parameter is q = (λ2−V2)1/2. This means that the point λ = 0 for the case of Yukawa potentials corresponds here to the point q = 0. We also find that the Regge trajectories λ (E) may have branch points of the square root type at finite, in general complex, values of E at which points the pole passes the origin λ = 0. We further find that the kinematic singularity of the S matrix at k = 0 is more complicated than it is for Yukawa potentials and is here characterized by the Floquet parameter ν (λ,k) associated with the Schrödinger equation. We illustrate these and other results with some