Dense Pure Point Spectrum on Quasi-Conical Domains

Definitions and mathematical statement

Let $\Omega\subset{\mathbb{R}}^d$ be an open connected set, with the ambient dimension $d$ left unspecified by the supplied record. Under the usual definition, $\Omega$ is quasi-conical if it contains Euclidean balls of arbitrarily large radius: \[ \forall R>0\;\exists x\in\Omega\quad B(x,R)\subset\Omega. \] The intended dimension and the compatibility of this definition with the report's construction require confirmation from the missing construction context.

The Dirichlet Laplacian $H_\Omega=-\Delta_\Omega^D$ is the nonnegative self-adjoint operator on $L^2(\Omega)$ associated with the closed quadratic form \[ q_\Omega[u]=\int_\Omega |\nabla u|^2\,{\,\mathrm{d}} x, \qquad \operatorname{dom}(q_\Omega)=H_0^1(\Omega), \] where $H_0^1(\Omega)$ is the closure of $C_c^\infty(\Omega)$ in the Sobolev $H^1$ norm. This specifies the Dirichlet boundary condition without assuming a smooth boundary. Write $\sigma_{\mathrm{ac}}(H_\Omega)$, $\sigma_{\mathrm{sc}}(H_\Omega)$, and $\sigma_{\mathrm p}(H_\Omega)$ for its absolutely continuous spectrum, singular continuous spectrum, and set of eigenvalues, respectively.

Intended problem, with unresolved construction data. In the particular construction of connected quasi-conical domains with adjustable hole sizes referred to by the supplied report passage, can the hole sizes be chosen so that \[ \sigma_{\mathrm{ac}}(H_\Omega)=\varnothing \qquad\text{and}\qquad \sigma_{\mathrm{sc}}(H_\Omega)=\varnothing? \] The requested pure point conclusion means that $L^2(\Omega)$ is the closed linear span of eigenfunctions of $H_\Omega$; the eigenvalues are then dense in the full spectrum: \[ \overline{\sigma_{\mathrm p}(H_\Omega)}=\sigma(H_\Omega). \] This does not require every point of the spectrum to be an eigenvalue.

The existential choice is restricted to the report's intended hole construction. The supplied record does not specify that construction, the holes' locations or shapes, the indexing of their sizes, or the admissible size constraints. Consequently, the displayed question preserves the spectral target but is not a complete specification of the family over which the existence quantifier ranges; it must not be read as replacing that family by all connected quasi-conical domains.

Short English statement

Choose the hole sizes within the report's intended connected quasi-conical construction so that the Dirichlet Laplacian has only pure point spectral type, with eigenvalues dense in its spectrum. The supplied record omits the construction and admissibility conditions needed to specify the choice precisely. Its open status is supplied dated curation, not an independently verified assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005613$, title Dense Pure Point Spectrum on Quasi-Conical Domains, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied. The registry gives proposed year $2023$.
• Supplied source citation: Geometric Spectral Theory (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/36, pp. 2017--2106. Source URL and supplied literature reference labelled "original OWR report": https://doi.org/10.4171/owr/2023/36. The record describes these reports as documenting mathematical workshops and their open-problem discussions. No passage-specific page or construction details are supplied.
• Supplied literature reference: Krejcirik--Lotoreichik, "quasi-conical domains": https://doi.org/10.1112/blms.13113. The supplied record does not provide a full bibliographic citation or identify a specific result from this reference.
• Literature assessment checked at 2026-08-22: supplied dated curation, not independently verified. The supplied assessment labels the problem open and states that the 2024 construction achieves empty absolutely continuous spectrum and embedded eigenvalues, while emptiness of singular continuous spectrum remains open. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and states that it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims; this draft does not independently verify them or claim to have consulted either external source.