Rellich uniqueness for the sound-soft spherical obstacle

By [F4], for every degree $\mathrm{\ell }$ the projection of $u$ has the form $u_{\mathrm{\ell }}(r,\cdot )=c_{\mathrm{\ell }}{h}_{\mathrm{\ell }}^{(1)}(kr)$ with $c_{\mathrm{\ell }}$ independent of $r$. Taking the degree-$\mathrm{\ell }$ projection of the zero trace at $r=a$ gives $0=c_{\mathrm{\ell }}{h}_{\mathrm{\ell }}^{(1)}(ka)$. The nonvanishing in [F3] implies $c_{\mathrm{\ell }}=0$ for every $\mathrm{\ell }$.

Thus every harmonic projection of $u(r,\cdot )$ vanishes for each $r>a$. By the uniform density in [F2], choose for each $\epsilon >0$ a finite sum $p$ of harmonic modes with $\mathrm{\parallel }u(r,\cdot )-p{\mathrm{\parallel }}_{\mathrm{\infty }}<\epsilon$. Orthogonality gives $⟨u,p⟩=0$, so $\mathrm{\parallel }u(r,\cdot ){\mathrm{\parallel }}_2^2=\mathrm{\mid }⟨u,u-p⟩\mathrm{\mid }\le \mathrm{\parallel }u(r,\cdot ){\mathrm{\parallel }}_2\mathrm{\parallel }u(r,\cdot )-p{\mathrm{\parallel }}_2\le \sqrt{4\pi }\epsilon \mathrm{\parallel }u(r,\cdot ){\mathrm{\parallel }}_2$. Letting $\epsilon \downarrow 0$ gives $u(r,\cdot )=0$ in $L^2$, and continuity makes it zero pointwise. Since $r>a$ was arbitrary, $u\equiv 0$.