Lemma: Positive-frequency scattering

For every fixed $\ell$ and $\omega >0$, the radial equation

has uniquely normalized Jost solutions $f_{+}$ and $f_{-}$ with

including the corresponding derivative asymptotics.

They construct a unitary scattering matrix $S_{\ell }(\omega )$. The horizon-to-infinity coefficient is

where

In particular,

All these coefficients are continuous on compact subsets of $(0,\infty )$.

Proof. The outgoing right Jost solution is constructed by the Volterra equation

Successive iteration converges absolutely. Indeed, the $n$-th iterated correction is bounded by

and consequently

The same Volterra estimate applied to a difference proves uniqueness. Differentiation of (eq:Volterra) gives (eq:ODE) and the derivative asymptotics at $+\infty$.

Expanding the sine in (eq:Volterra) gives, at the other end,

The omitted tails tend to zero because $V_{\ell }{f}_{+}\in L^1$. Differentiating the same expressions gives the left derivative asymptotics.

For every solution of (eq:ODE), the current

is constant. For $f_{+}$, its right-end value is $\omega$, whereas its left-end value is

Thus $a_{\ell }\ne 0$, and the horizon-incident solution $q_{\mathrm{H}}=f_{+}/a_{\ell }$ has asymptotics

This proves the transmission formula and its bounds.

The left Jost solution is obtained from the analogous Volterra equation integrated from $-\infty$. Its right incoming coefficient is likewise nonzero by current conservation, so it constructs the solution $q_{\mathrm{I}}$ incident from infinity.

For arbitrary incoming amplitudes $A_{\mathrm{H}},A_{\mathrm{I}}$, the corresponding combination of $q_{\mathrm{H}}$ and $q_{\mathrm{I}}$ has outgoing amplitudes

Current conservation gives

for every pair of incoming amplitudes. Therefore $S_{\ell }^{\ast }{S}_{\ell }=I$, and the square matrix $S_{\ell }$ is unitary.

On a compact positive-frequency interval, the Volterra series have uniform majorants. Their terms and the coefficient integrals depend continuously on $\omega$. Dominated convergence proves the last assertion.