Asymptotic Form of theSMatrix for Large Angular Momentum in the Left Half-Plane

Starting with the Schrödinger equation, we prove that for all energies, $S(\lambda , s)$ approaches $e^{2i\lambda \pi }$ as $\left|\lambda \right|$ becomes large in the direction $\frac{1}{2}\pi <\arg \lambda <\frac{3}{2}\pi$, for a class of potentials. These include the square-well potential, the cut-off Coulomb potential, a single Yukawa potential, and a superposition of Yukawa potentials of the form $\int {\mu }^{\infty }\left(\frac{e^{-{\mu }^{\prime }r}}{r}\right)e^{-{\mu }^{\prime }}d{\mu }^{\prime }$. The asymptotic forms of the Regge-pole parameters ${\alpha }_n$ and ${\beta }_n$ are derived. We found that $\arg {\lambda }_n$ approaches $\frac{1}{2}\pi$ or $\frac{3}{2}\pi$ as $n\to \infty$, and ${\beta }_n$ is proportional to $1+e^{2\pi i{\alpha }_n}$, which grows exponentially for the Regge poles in the lower half plane. The asymptotic forms for the Jost functions and the $Y$ function are also given. A general proof for the asymptotic formula $S(\lambda , s)\to e^{2\pi i\lambda }$ as $\left|\lambda \right|\to \infty$, $\frac{1}{2}\pi <\arg \lambda <\frac{3}{2}\pi$, is also outlined.