Lemma: Zero-energy normalization

The potential $V$ is smooth, nonnegative, and satisfies

Its zero-energy Jost solutions are

and

There is an ${\omega }_c>0$ for which

Proof. Since $\mathrm{d}x=F^{-1}\mathrm{d}r$,

Near the horizon, $x$ grows logarithmically in $r-1$; at infinity, $x=r+O(\log r)$. These facts prove the first-moment bound.

Substitution in $\mathsf{L}f=0$ verifies the two solutions. They tend to one at their respective normalization ends. Using ${\partial }_{x}r=F$ and $F^{\prime }=r^{-2}$ gives their Wronskian $-1$.

The Jost Volterra equations have the zero-frequency kernel $y-x$ in the right equation and its left-end analogue. The first-moment bound gives convergent series and continuity at $\omega =0$, including derivatives on compact $x$-sets. The nonzero Wronskian therefore gives the final assertion.