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 L² 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 integer d ≥ 2, the theorem constructs some connected enlarging-cube tower with jointly adaptive room sizes and positive vanishing windows. The ordinary Dirichlet Laplacian has a complete orthonormal eigenbasis and spectrum and essential spectrum [0, ∞); both continuous spectral subspaces vanish. The eigenvalues are positive, countable and dense in the half-line. The spectral conclusion concerns pure-point spectral type. The squared L2 tail bound 4^(-n) is indexed by rooms. One prescribed positive eigenvalue can be included by dilation.
Disclosures
AI tools assisted research, proof development, writing, and verification. The cube/window architecture and local domain perturbation tools are credited to the cited predecessors.
Reproduction
Manuscript version 1.0, October 20, 2026. The six-member package contains the complete six-page manuscript, LaTeX source and bibliography, a portable Python build and integrity helper, README, citation metadata and checksum manifest. The proof is entirely analytic. Python 3.9 or later and an installed Tectonic, latexmk or pdfLaTeX can rebuild the source using the README instructions and standard TeX packages. The helper installs no software; offline Tectonic requires cached resources. Compiler versions can change PDF bytes while preserving content. The checksum check verifies the five listed files, not mathematical correctness.
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.