Absence of singularly continuous spectrum in the embedded-eigenvalue construction

Determine whether the connected quasi-conical domain constructed by Krejčiřík and Lotoreichik, with sufficiently small windows between adjacent cubes, can be chosen so that the singularly continuous spectrum of its Dirichlet Laplacian is empty.

Background

The paper discusses a connected quasi-conical domain whose Dirichlet Laplacian has prescribed embedded eigenvalues and whose absolutely continuous spectrum is empty. The remaining unresolved spectral component is the singularly continuous spectrum. The problem asks whether the window sizes in that construction can be selected to eliminate this component as well, which would produce a connected quasi-conical domain with purely point spectrum densely filling the non-negative half-line.

References

We leave as an open problem whether the singularly continuous spectrum is empty as well. In the example Ω of (see Figure~\ref{Fig.embedded}), is it possible to select the sizes of the windows so small that $$ \sigma_\mathrm{sc}(-\Delta_D\Omega) = \varnothing \,? $$

— Spectral geometry: old questions and new answers  (2609.28602 - Krejcirik, 23 Sep 2026) in Open Problem following Theorem 2.4, Section 2.3 (page unavailable in source)