Dimension of null and defect subspaces for higher-order s-points

Determine the dimensions of the null-control subspace \(\mathscr N\) and the unreachable-state subspace \(\mathscr D\) for s-points of order \(m>1\) in the three-dimensional acoustic scattering system with compactly supported potential.

Background

The paper defines an s-point as a refocusing point at which the shifted-potential acoustic scattering system is not controllable. S-points have an order determined by the smallest positive integer mm for which a finite-energy state satisfies a q-polyharmonic equation of order mm. For an uncontrollable system, N=ker⁡W\mathscr N=\ker W consists of null-controls and D=H⊖U\mathscr D=\mathscr H\ominus\mathscr U consists of unreachable states; Fredholm theory establishes that these spaces have equal finite dimension. The paper characterizes the dimension in the unique-negative-level case for first-order s-points, but leaves the corresponding dimensions for higher-order s-points unresolved.

References

What is the dimension of the subspaces $\mathscr N$ and $\mathscr D$ for the s-points of the order $m>1$?

— Controllability, returning waves and scattering without reverberation in 3D acoustic dynamic system  (2609.03536 - Belishev et al., 3 Sep 2026) in Comments, bullet beginning “What is the dimension...”

Is it possible to extend our results to a system described by the equation $\rho u_{tt}-\Delta_gu+qu=0$ with a density $\rho>0$ and metric $g$, provided that $\rho\equiv 1$, $g_{ij}\equiv\delta_{ij}$, and $q\equiv 0$ holds as $|x|>R_*$? The difficulty is that not every smooth locally perturbed Euclidean metric allows one to focus incoming waves on an arbitrarily given point footnote{S.V.Ivanov, private communication}. As we guess and hope, the latter is possible for a metric $g$ that is sufficiently close to the Euclidean one everywhere.

— Controllability, returning waves and scattering without reverberation in 3D acoustic dynamic system  (2609.03536 - Belishev et al., 3 Sep 2026) in Comments, bullet beginning “Is it possible to extend our results...”

Of particular interest is the modeling of null-controls and the corresponding r-waves, but it has not yet been possible to find any explicit representations for them suitable for numerical implementation. The question comes down to whether equations of the type (\ref{Eq for Phi}) can be solved efficiently.

— Controllability, returning waves and scattering without reverberation in 3D acoustic dynamic system  (2609.03536 - Belishev et al., 3 Sep 2026) in Example, final bullet before Appendix