Preprint Mathematics
Pure-Point Dirichlet Spectrum in an Enlarging Cube Tower
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Overview
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.
Keywords Dirichlet Laplacian · quasi-conical domain · pure-point spectral type · embedded eigenvalue · completeness · domain perturbation
Scope of the work
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. Pure-point spectral type does not mean every spectral point is an eigenvalue or that the resolvent is compact. The squared L² tail bound 4^(−n) is indexed by rooms, not Euclidean distance. One prescribed positive eigenvalue can be included by dilation. Arbitrary preassigned towers, a prescribed full eigenvalue sequence, effective numerical apertures, simplicity and physical-distance decay rates are not asserted.
Reproduction notes
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; no numerical experiment is a proof premise. 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.
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Scott Schubr. (2026). Pure-Point Dirichlet Spectrum in an Enlarging Cube Tower (Version 1). Luminary Research Archive. https://luminaryoddities.com/researcharchive/pure-point-dirichlet-spectrum-in-an-enlarging-cube-tower/v/1
BibTeX
@techreport{luminary_pure_point_dirichlet_spectrum_in_an_enlarging_cube_tower_v1,
title = {Pure-Point Dirichlet Spectrum in an Enlarging Cube Tower},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Preprint},
note = {Version 1},
url = {https://luminaryoddities.com/researcharchive/pure-point-dirichlet-spectrum-in-an-enlarging-cube-tower/v/1}
}Reader correspondence
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