Exact-DtN truncated modal variational formulation

Fix $0<a<R$ and $k>0$. The coefficient energy space ${\mathcal{V}}_{a,R}$ consists of sequences $u=(u_{\mathrm{\ell }}{)}_{\mathrm{\ell }\ge 0}$ with $u_{\mathrm{\ell }}\in H^1((a,R);{\mathcal{H}}_{\mathrm{\ell }})$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathcal{V}}^2={\sum }_{\mathrm{\ell }=0}^{\mathrm{\infty }}{\int }_a^R(r^2\mathrm{\parallel }{u}_{\mathrm{\ell }}^{\mathrm{\prime }}(r){\mathrm{\parallel }}_2^2+(1+\mathrm{\ell }(\mathrm{\ell }+1))\mathrm{\parallel }{u}_{\mathrm{\ell }}(r){\mathrm{\parallel }}_2^2)dr<\mathrm{\infty }\mathrm{.}$ For smooth inner data $f$ on ${\mathbb{S}}^2$, let $f_{\mathrm{\ell }}$ be its projection onto ${\mathcal{H}}_{\mathrm{\ell }}$ and define the affine trial space ${\mathcal{V}}_f=\{u\in {\mathcal{V}}_{a,R}:u_{\mathrm{\ell }}(a)=f_{\mathrm{\ell }}\text{ for every }\mathrm{\ell }\}$ and test space ${\mathcal{V}}_0=\{v\in {\mathcal{V}}_{a,R}:v_{\mathrm{\ell }}(a)=0\text{ for every }\mathrm{\ell }\}$. Traces and continuity of the following form are established in Modal trace bound and continuity of the exact-DtN variational form ↗. With the inner product linear in its first argument, set ${\mathfrak{a}}_{k,a,R}(u,v)={\sum }_{\mathrm{\ell }=0}^{\mathrm{\infty }}{\int }_a^R(r^2⟨u_{\mathrm{\ell }}^{\mathrm{\prime }},v_{\mathrm{\ell }}^{\mathrm{\prime }}⟩+(\mathrm{\ell }(\mathrm{\ell }+1)-k^2{r}^2)⟨u_{\mathrm{\ell }},v_{\mathrm{\ell }}⟩)dr-R^2{\sum }_{\mathrm{\ell }=0}^{\mathrm{\infty }}{\tau }_{\mathrm{\ell }}(k,R)⟨u_{\mathrm{\ell }}(R),v_{\mathrm{\ell }}(R)⟩\mathrm{.}$ The exact-DtN variational problem is to find $u\in {\mathcal{V}}_f$ such that ${\mathfrak{a}}_{k,a,R}(u,v)=0$ for every $v\in {\mathcal{V}}_0$. The assigned theorem on this page proves existence and uniqueness.