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 $\frac{1}{r}$. Then at large energy (positive or negative) there will be Regge poles at all negative integral values of $l$. If the $\frac{1}{r}$ term is absent in the expansion of the potential, the pole at $l=-1\mathrm{}$ will be absent. Generally, if $r^{2n-1}$ is the first nonvanishing odd power term in the potential expansion, the poles at $-1, -2, \cdots , -n$ 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 $-\frac{a}{r^2}$ at small $r$, the scattering amplitude has a cut in the $l$ plane from $-\frac{1}{2}-a^{\frac{1}{2}}$ to $-\frac{1}{2}+a^{\frac{1}{2}}$, 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 $-l-1\mathrm{}$ but that the residues at these poles are, in general, different from each other.