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
- 1 · 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 has a complete orthonormal eigenbasis and essential spectrum [0,infinity). 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),sqrt(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, computable 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. Crack-aware primal and flux approximation certify whole-resolvent compressions and strict whole-cluster spectral tests. Explicit sufficient aperture bounds are logarithmic in the plane and linear in higher dimensions. For uniformly diverging spatial weights, a uniformly bounded all-energy 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
The geometry is selected after the spatial weight and spectral filter or energy rate. Effective finite-prefix construction assumes rational initial geometry, a supplied total computable rational spatial envelope and a true computable filter-tail or rate-divergence modulus. Bounds concern operator norms for bounded source regions; constants can depend on the source. No runtime estimate or practical aperture-size claim is made.
Reproduction notes
The eleven-member package contains the complete eighteen-page PDF, four self-contained LaTeX sources, two portable build/integrity and exact finite-arithmetic scripts, expected arithmetic data, reader instructions, citation metadata and checksums. Analytic proofs establish the general filter/rate and termination theorems; the arithmetic checker does not implement a numerical PDE solver.
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Cite this edition
Scott Schubr. (2026). Energy-Filtered Dynamical Localization in Constructive Dirichlet Towers (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/energy-filtered-dynamical-localization-in-constructive-dirichlet-towers/v/1
BibTeX
@techreport{luminary_energy_filtered_dynamical_localization_in_constructive_dirichlet_towers_v1,
title = {Energy-Filtered Dynamical Localization in Constructive Dirichlet Towers},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/energy-filtered-dynamical-localization-in-constructive-dirichlet-towers/v/1}
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