Behavior of Regge Poles in a Potential at Large Energy

Following Mandelstam's suggestion, we consider a potential which can be expanded in a power series in r, beginning with 1/r. Then at large energy (positive or negative) there will be Regge poles at all negative integral values of l. If the 1/r term is absent in the expansion of the potential, the pole at l=-1 will be absent. Generally, if r^2n-1 is the first nonvanishing odd power term in the potential expansion, the poles at -1, -2, ⋯, -n are absent. If the potential expansion contains only even non-negative powers of r, there will be no Regge poles at finite negative l in the limit of large energy. If the potential behaves as -ar² at small r, the scattering amplitude has a cut in the l plane from -1/2-a1/2 to -1/2+a1/2, and the Regge poles are no longer located at negative integers at large energy. For finite energy, it is shown that a pole at a positive integer or half-integer l implies another pole at -l-1 but that the residues at these poles are, in general, different from each other.