Pure-Point Dirichlet Spectrum in an Enlarging Cube Tower
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Abstract
For every dimension d≥2, we construct a connected quasi-conical open set whose Dirichlet Laplacian has a complete orthonormal basis of eigenfunctions and spectrum [0,∞). The domain is a tower of adjacent enlarging cubes joined through positive windows tending to zero. Cube sizes and windows are chosen together. The construction gives an affirmative existence answer to the singular-continuous-spectrum question raised for this geometry by Krejčiřík and Lotoreichik and repeated in a recent spectral-geometry survey. At each finite attachment we continue a nested family of whole spectral clusters, including their multiplicities. A summable projection budget and a refreshed dense family of test vectors make the limiting finite-dimensional reducing subspaces complete. We also obtain a room-index L^2 tail bound and retain the possibility of placing one prescribed positive embedded eigenvalue by dilation.
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Research details
- Subjects
- Mathematics · Analysis and Spectral Theory
- Keywords
- Dirichlet Laplacian, quasi-conical domain, pure-point spectral type, embedded eigenvalue, completeness, domain perturbation
- Manuscript date
- 2026-10-20
- Reader reuse
- No additional permission granted
Scope
For every d≥2 an adaptively chosen connected enlarging-cube tower has a complete Dirichlet eigenfunction basis and spectrum [0,∞), with room-index L² tail control and any one prescribed positive eigenvalue obtainable by dilation.
Disclosures
AI tools assisted research, proof development, writing, and verification. This author-published preprint is not external human refereeing, journal acceptance or formal verification. The cube/window architecture and local domain perturbation tools are credited to the cited predecessors; worldwide priority remains provisional.
Reproduction
The 6-page manuscript and complete editable source are supplied in this 6-member package. README.md gives LaTeX build instructions. build_paper.py supports installed Tectonic, latexmk or pdfLaTeX and checks SHA256SUMS.txt. The existence construction and spectral conclusions are proved analytically; room sizes and windows are selected together.
References and related resources
- D. Daners, Domain perturbation for linear and semi-linear boundary value problems, in M. Chipot (ed.), Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 6, Elsevier, 2008, pp. 1–81.
- D. Krejčiřík and V. Lotoreichik, Quasi-conical domains with embedded eigenvalues, Bulletin of the London Mathematical Society 56 (2024), 2969–2981. Preprint: arXiv:2205.08172v2.
- D. Krejčiřík, Spectral geometry: old questions and new answers, arXiv:2609.28602v1, 23 September 2026.