Existence and uniqueness for the spherical exact-DtN variational problem

By [F3] applied at the inner radius $a$, the proposed series has inner trace $f$. Its degree-$\mathrm{\ell }$ radial ratio has modulus at most one for $r\ge a$ by [F5], and its radial derivative is bounded by $k+(\mathrm{\ell }+1)\mathrm{/}r$ times that ratio. The angular energy of degree $\mathrm{\ell }$ is $\mathrm{\ell }(\mathrm{\ell }+1)$ times its squared $L^2$ norm. Consequently its total $\mathcal{V}$ energy is at most $C_{a,R,k}{\sum }_{\mathrm{\ell }}(1+\mathrm{\ell }{)}^2\mathrm{\parallel }{f}_{\mathrm{\ell }}{\mathrm{\parallel }}_2^2$, which is finite by [F4]. Thus it belongs to ${\mathcal{V}}_f$.

Let $w\in {\mathcal{V}}_0$ solve the homogeneous variational equation. Testing against functions supported in a single degree and compactly supported in $(a,R)$ shows distributionally that $(r^2{w}_{\mathrm{\ell }}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}+(k^2{r}^2-\mathrm{\ell }(\mathrm{\ell }+1))w_{\mathrm{\ell }}=0$. Since the coefficients are smooth and $r>0$, this one-dimensional equation implies $w_{\mathrm{\ell }}\in C^2([a,R];{\mathcal{H}}_{\mathrm{\ell }})$. Integrating by parts and allowing arbitrary outer test trace yields $w_{\mathrm{\ell }}^{\mathrm{\prime }}(R)={\tau }_{\mathrm{\ell }}{w}_{\mathrm{\ell }}(R)$; the inner trace is $w_{\mathrm{\ell }}(a)=0$.

Each modal coefficient satisfies $(r^2{u}_{\mathrm{\ell }}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}+(k^2{r}^2-\mathrm{\ell }(\mathrm{\ell }+1))u_{\mathrm{\ell }}=0$ and $u_{\mathrm{\ell }}^{\mathrm{\prime }}(R)={\tau }_{\mathrm{\ell }}{u}_{\mathrm{\ell }}(R)$ by [F3]. Integration by parts against a finite-degree test with zero inner trace makes its modal volume term equal $R^2{u}_{\mathrm{\ell }}^{\mathrm{\prime }}(R)\overline{v_{\mathrm{\ell }}(R)}$; the outer term in [F1] subtracts exactly $R^2{\tau }_{\mathrm{\ell }}{u}_{\mathrm{\ell }}(R)\overline{v_{\mathrm{\ell }}(R)}$. Hence the form vanishes for every finite-degree test. Finite-degree truncations converge in $\mathcal{V}$, and [F2] extends the identity to every test in ${\mathcal{V}}_0$. This proves existence.

The radial ODE solution is $w_{\mathrm{\ell }}(r)=A_{\mathrm{\ell }}{h}_{\mathrm{\ell }}^{(1)}(kr)+B_{\mathrm{\ell }}{h}_{\mathrm{\ell }}^{(2)}(kr)$: the two solutions have nonzero Wronskian by [F5], and matching value and derivative at one radius followed by uniqueness of the linear initial-value equation spans all solutions. At $R$, the condition from step 1.2 gives $B_{\mathrm{\ell }}k((h_{\mathrm{\ell }}^{(2)}{)}^{\mathrm{\prime }}(kR)-[(h_{\mathrm{\ell }}^{(1)}{)}^{\mathrm{\prime }}(kR)\mathrm{/}{h}_{\mathrm{\ell }}^{(1)}(kR)]h_{\mathrm{\ell }}^{(2)}(kR))=0$. The parenthesized factor is the nonzero Wronskian divided by $h_{\mathrm{\ell }}^{(1)}(kR)$, so $B_{\mathrm{\ell }}=0$. The inner condition and [F5] then give $A_{\mathrm{\ell }}=0$. Every modal projection vanishes, hence $w=0$ in $\mathcal{V}$. This proves uniqueness.