Radial volume scattering separates into matched partial-wave boundary problems

Assume Countable Choice and Dependent Choice. Let $R>0$, $k>0$, and $Q\in C([0,R];\mathbb{C})$. Set $q(x)=Q(\mathrm{\mid }x\mathrm{\mid })$ on $B_R$ and $q(x)=0$ outside $B_R$. Choose for every $\mathrm{\ell }$ an orthonormal basis $Y_{\mathrm{\ell }m}$ of ${\mathcal{H}}_{\mathrm{\ell }}$. Suppose $u^{\mathrm{i}\mathrm{n}\mathrm{c}}$ is a classical free Helmholtz field on all of ${\mathbb{R}}^3$, and $u=u^{\mathrm{i}\mathrm{n}\mathrm{c}}+V$ satisfies the volume-potential equation $V(x)={\int }_{B_R}{G}_k(x-y)q(y)u(y)dy$. Assume $u$ is continuous on ${\bar{B}}_R$, piecewise $C^2$ on either side of $\mathrm{\mid }x\mathrm{\mid }=R$, and $u$ and its radial derivative are continuous across that interface. For every spherical harmonic $Y_{\mathrm{\ell }m}$ assume its coefficient $w_{\mathrm{\ell }m}(r):=r{\int }_{S^2}u(r\omega )\overline{Y_{\mathrm{\ell }m}(\omega )}d\sigma (\omega )$ satisfies the regular-origin condition $w_{\mathrm{\ell }m}(r)=O(r^{\mathrm{\ell }+1})$ as $r\downarrow 0$. Explicitly assume the incident regular-mode form: for every $\mathrm{\ell },m$ there is ${\alpha }_{\mathrm{\ell }m}\in \mathbb{C}$ such that $r{\int }_{S^2}{u}^{\mathrm{i}\mathrm{n}\mathrm{c}}(r\omega )\overline{Y_{\mathrm{\ell }m}(\omega )}d\sigma (\omega )={\alpha }_{\mathrm{\ell }m}r{j}_{\mathrm{\ell }}(kr)$ for $r>0$. The outgoing radiation condition is ${\lim }_{r\to \mathrm{\infty }}{\sup }_{\omega \in S^2}\mathrm{\mid }r({\mathrm{\partial }}_{r}V(r\omega )-ikV(r\omega ))\mathrm{\mid }=0$. Then each reduced mode satisfies $w_{\mathrm{\ell }m}^{\mathrm{\prime }\mathrm{\prime }}(r)+(k^2+Q(r)-\frac{\mathrm{\ell }(\mathrm{\ell }+1)}{r^2})w_{\mathrm{\ell }m}(r)=0(0<r<R),$ is regular at $0$, and has for $r>R$ the form $w_{\mathrm{\ell }m}(r)={\alpha }_{\mathrm{\ell }m}r{j}_{\mathrm{\ell }}(kr)+{\beta }_{\mathrm{\ell }m}r{h}_{\mathrm{\ell }}^{(1)}(kr)\mathrm{.}$ The interior and exterior values and first derivatives agree at $R$. This is a conditional mode reduction and matching statement; it asserts neither existence nor uniqueness of the total field.

For a fixed finite set of orders and a positive matching radius, Soundness and eventual absolute precision for spherical Bessel enclosures supplies certified scaled values and argument derivatives of $j_{\mathrm{\ell }}$ and outgoing $h_{\mathrm{\ell }}^{(1)}$ on positive real argument enclosures. Scaled spherical Bessel evaluation on positive real arguments also permits requesting their unscaled counterparts. At $z=kR>0$, the exterior matching columns are $R{j}_{\mathrm{\ell }}(z)$ and $R{h}_{\mathrm{\ell }}^{(1)}(z)$, with radial derivatives $j_{\mathrm{\ell }}(z)+z{j}_{\mathrm{\ell }}^{\mathrm{\prime }}(z)$ and $h_{\mathrm{\ell }}^{(1)}(z)+z(h_{\mathrm{\ell }}^{(1)}{)}^{\mathrm{\prime }}(z)$. These identities follow by the product and chain rules. Thus the supplied first argument derivatives suffice, including at $\mathrm{\ell }=0$.

The real-argument interface does not restrict the theorem's complex contrast $Q$: $Q$ enters the interior ODE while the exterior free wave number $k$ remains positive real. No evaluation at a complex wave number is used here. Certified exterior matching columns do not certify integration of the interior ODE, solvability of the complete matching system, total-field uniqueness, or the angular cutoff. Those claims retain their separate hypotheses and proofs. The mathematical mode-reduction statement above remains valid independently of whether numerical parameter enclosures have been provided.

For a numerical evaluation of these columns, supplied terminating positive-real parameter interfaces for $k$ and $R$ induce an argument interface for $kR$. On a width request $2^{-n}$, successively request both parameter boxes at indices $q=n,n+1,\dots$, skip until both lower endpoints are positive, multiply their dyadic intervals exactly, and stop when the product width is at most $2^{-n}$. Each call terminates by hypothesis; positivity holds eventually because $k,R>0$. The parameter boxes are eventually bounded, and the product-width bound $\mathrm{width}(XY)\le \sup \mathrm{\mid }X\mathrm{\mid }\mathrm{width}(Y)+\sup \mathrm{\mid }Y\mathrm{\mid }\mathrm{width}(X)$ tends to zero. Thus the scan terminates for every requested $n$ and retains the exact, generally non-dyadic product. The same interval operations propagate the factors $R$ and $z=kR$ in the displayed matching columns. This numerical representation condition does not restrict the theorem's exact real parameters.