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

Starting with the Schr\"odinger equation, we prove that for all energies, S(λ, s) approaches e^2iλπ as |λ| becomes large in the direction 1/2π<argλ<3/2π, 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 ${∫}{{μ}}^{{∞}}({{e}^{{-}{{μ}}^{{'}}r}}{r}){e}^{{-}{{μ}}^{{'}}}d{{μ}}^{{'}}$. The asymptotic forms of the Regge-pole parameters ${{α}}ₙ$ and ${{β}}ₙ$ are derived. We found that $arg{{λ}}ₙ$ approaches $1/2{π}$ or $3/2{π}$ as $n{→}{∞}$, and ${{β}}ₙ$ is proportional to $1+{e}^{2{π}i{{α}}ₙ}$, 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 $S({λ}, s){→}{e}^{2{π}i{λ}}$ as $|{λ}|{→}{∞}$, $1/2{π}<arg{λ}<3/2{π}$, is also outlined.