On controllability of the acoustic scattering dynamical system in $\Bbb R^3$

The acoustic scattering problem is to find $u=u^f(x,t)$ satisfying
\begin{align*} &u_{tt}-\Delta u+qu=0, (x,t) \in {\mathbb R}^3 \times (-\infty,0);\\ &u \mid_{|x|<-t} =0 , t<0\\ &\lim_{s \to \infty} s u((s+\tau) \omega,-s)=f(\tau,\omega), (\tau,\omega) \in \Sigma:=[0,\infty)\times S^2; \end{align*}
with a real valued compactly supported potential $q\in L_\infty(\Bbb R^3)$ and a control $f \in \mathscr F:=L_2(\Sigma)$. Let $\mathscr F^\xi:= \{f\in\mathscr F | f\big|_{0\leqslant \tau\leqslant \xi}=0\}$, $\mathscr H:=L_2(\Bbb R^3)$, $\mathscr H^\xi:=\{y\in \mathscr H | y\big|_{|x|<\xi}=0\}$, $\xi>0$. For the (delayed) controls $f\in\mathscr F^\xi$, the reachable set is $\mathscr U^\xi:=\{u^f(\cdot, 0) | f\in\mathscr F^\xi\}\subset\mathscr H^\xi$, whereas $\mathscr D^\xi:=\mathscr H^\xi\ominus\mathscr U^\xi$ is the defect (unreachable) subspace. The paper provides a characterization of $\mathscr D^\xi$ as follows.
We say an $a\in\mathscr H^\xi$ to be a $q$-polyharmonic function of the order $n$ if $(-\Delta +q)^n a=0$ holds for $|x|>\xi$, and write $a\in\mathscr A^\xi_n$. Our main result is the relation
\begin{equation*} {\mathscr D}^\xi =\overline{{\rm span }\{\mathscr A^\xi_n | n\geqslant 1\}}, \xi>0 \end{equation*}
(the closure in $\mathscr H$). It basically concludes the study of controllability of the acoustical dynamical system governed by the locally perturbed wave equation in $\mathbb R^3$.