Research Article · Preprint Mathematics
Energy-Filtered Dynamical Localization in Constructive Dirichlet Towers
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
For every dimension d≥2 and every measurable spatial weight w≥1 essentially bounded on bounded sets, we construct a connected tower of enlarging cubes whose ordinary Dirichlet Laplacian H has a complete orthonormal eigenbasis and essential spectrum [0,∞). Given any finite nondecreasing r(E)≥1 tending to infinity, the tower can be chosen so that weighted local-source evolution below energy E is bounded uniformly in time by C_K min{r(E),√(E+1)}. More generally, any prescribed Borel spectral filter 0<h≤1 uniformly vanishing at high energy admits a uniform weighted local-source bound. Local Neumann approximation targets and a summable spectral-projection budget supply the uniform source estimate. With a rational initial room and a supplied computable sequence of local weight upper bounds and a supplied computable filter-tail modulus, the geometry and its finite spectral certificates can be produced by a procedure terminating at every stage. We prove this using crack-aware conforming primal and flux approximation, certified whole-resolvent compression and strict whole-cluster spectral tests. An explicit logarithmic aperture bound applies in the plane; in higher dimensions a heat-semigroup argument gives a linear sufficient bound. We also prove that the corresponding unfiltered all-energy bound fails for uniformly diverging weights. Thus an arbitrarily slow divergent energy factor can be prescribed, whereas a uniformly bounded factor is impossible. The geometry depends on the prescribed rate.
Keywords Dirichlet Laplacian · dynamical localization · quasi-conical domain · spectral approximation · constructive geometry
Scope of the work
Connected Dirichlet towers in every dimension at least two with complete pure-point spectral type, essential spectrum [0,∞) and prescribed-filter weighted local-source bounds; computable input moduli yield a terminating finite-stage construction, while uniformly diverging weights obstruct a bounded all-energy factor.
Reproduction notes
The paper has 18 pages. The editable-source archive contains 11 files. Compile main.tex with its three included section files. Run standard-library verify_certificates.py with --check for the explicit finite arithmetic; build_paper.py also supports package-identity checks.
The manuscript
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Cite this edition
Scott Schubr. (2026). Energy-Filtered Dynamical Localization in Constructive Dirichlet Towers (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/energy-filtered-dynamical-localization-in-constructive-dirichlet-towers/v/2
BibTeX
@techreport{luminary_energy_filtered_dynamical_localization_in_constructive_dirichlet_towers_v2,
title = {Energy-Filtered Dynamical Localization in Constructive Dirichlet Towers},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/energy-filtered-dynamical-localization-in-constructive-dirichlet-towers/v/2}
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