Every outgoing spherical Helmholtz mode is an outgoing Hankel mode

Fix $\mathrm{\ell }$ and any $v\in {\mathcal{H}}_{\mathrm{\ell }}$. Define $w_v(r)=⟨u(r,\cdot ),v{⟩}_{L^2({\mathbb{S}}^2)}$. Integrating the polar-coordinate Helmholtz equation against $v$, using ${\mathrm{\Delta }}_{{\mathbb{S}}^2}v=-\mathrm{\ell }(\mathrm{\ell }+1)v$ and the angular integration-by-parts identity from [F2], gives $w_v^{\mathrm{\prime }\mathrm{\prime }}+2r^{-1}{w}_v^{\mathrm{\prime }}+(k^2-\mathrm{\ell }(\mathrm{\ell }+1)r^{-2})w_v=0$. Since this holds for every vector in the finite-dimensional space, the projected vector $u_{\mathrm{\ell }}(r,\cdot )$ solves the same scalar radial equation componentwise.

The two functions $h_{\mathrm{\ell }}^{(1)}(kr)$ and $h_{\mathrm{\ell }}^{(2)}(kr)$ solve this equation and have nonzero Wronskian by [F3]–[F4]; hence at any fixed $r_0>a$ their value-derivative vectors form an invertible $2\times 2$ matrix. Solve this matrix equation to match a coefficient and its derivative at $r_0$. The difference then solves a first-order system $Y^{\mathrm{\prime }}=M(r)Y$ with continuous $M$ and zero initial data. On each compact interval where $\mathrm{\parallel }M\mathrm{\parallel }\le C$, integration gives $\mathrm{\mid }Y(r)\mathrm{\mid }\le C{\int }_{r_0}^r\mathrm{\mid }Y(s)\mathrm{\mid }ds$ (or the reversed integral for $r<r_0$); Grönwall's inequality gives $Y=0$. Extending across compact intervals proves every coefficient is a linear combination of the two modes.

Apply the uniform radiation condition [F1] to any $v\in {\mathcal{H}}_{\mathrm{\ell }}$ and integrate it against $\bar{v}$; its coefficient tends to zero because its magnitude is bounded by the uniform supremum times $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^1}$. At $\mathrm{\ell }=0$, direct differentiation of $h_0^{(1)}(z)=-i{e}^{iz}\mathrm{/}z$ gives $(h_0^{(1)}{)}^{\mathrm{\prime }}=(i-z^{-1})h_0^{(1)}$, and conjugation gives the corresponding incoming formula. For $\mathrm{\ell }\ge 1$ use the derivative recurrence in [F3]. In both cases the asymptotic gives $r({\mathrm{\partial }}_r-ik)h_{\mathrm{\ell }}^{(1)}(kr)\to 0$. For the incoming conjugate, the same recurrence gives $r({\mathrm{\partial }}_r-ik)h_{\mathrm{\ell }}^{(2)}(kr)=-2i\overline{(-i{)}^{\mathrm{\ell }+1}}{e}^{-ikr}+o(1)$, whose magnitude tends to $2$. Thus its coefficient in the mode combination from step 1.2 must vanish. Since this holds for every $v$, the projected vector is $c_{\mathrm{\ell }}{h}_{\mathrm{\ell }}^{(1)}(kr)$ for a fixed $c_{\mathrm{\ell }}\in {\mathcal{H}}_{\mathrm{\ell }}$. The argument treats each degree separately and uses no interchange of the degree sum with the limit $r\to \mathrm{\infty }$.