Lemma: S-wave threshold bound

The potential is smooth and nonnegative,

The zero-energy Jost solutions are

Choose ${\omega }_c>0$ with $|\mathcal{W}(\omega )|\ge 1/2$ on $[0,{\omega }_c]$. For every $X>0$, a finite constant $C_X$ exists with

One explicit choice is

Proof. Use $\mathrm{d}x={\mathsf{f}}^{-1}\mathrm{d}r$ to obtain $\int V\mathrm{d}x={{\int }_1}^{\infty }{r}^{-3}\mathrm{d}r=1/2$. The first moment follows from logarithmic horizon behavior and $x=r+O(\log r)$ at infinity.

Direct substitution gives the zero-energy solutions and their Wronskian. The first-moment condition makes the Jost Volterra series continuous at zero, so its nonzero Wronskian supplies ${\omega }_c$.

In the low-frequency Volterra series use $|\sin (\omega z)/\omega |\le |z|$. This gives $|f_{\pm }|\le e^{m_X}$ on the compact region; $|\tau |\le 4\omega$ gives the threshold bound. For $\omega \ge {\omega }_c$, the series is bounded by $e^{\Vert V{\Vert }_1/{\omega }_c}$, and current conservation gives $|\tau |\le 1$.