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Energy-Filtered Dynamical Localization in Constructive Dirichlet Towers

Scott Schubr

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Abstract

We construct connected quasi-conical Euclidean domains whose Dirichlet evolution satisfies prescribed weighted bounds uniformly over the whole space of inputs supported in a fixed bounded region. For every dimension d≥2, every locally essentially bounded measurable weight w≥1, and every prescribed positive Borel spectral filter h≤1 uniformly vanishing at high energy, one enlarging-cube tower has a complete eigenbasis, essential spectrum [0,∞), and a weighted operator bound uniform over all bounded spectral multipliers of norm at most one. Consequently, the dependence on a finite energy ceiling can be bounded by any prescribed nondecreasing divergent rate, while a bounded all-energy factor is impossible for uniformly diverging weights. A logarithmic rate gives every positive fractional smoothing order on the same domain. The domain is chosen after the weight and filter. With a rational first room, computable local weight upper bounds and a computable filter-tail modulus, a deterministic procedure produces every finite prefix of the geometry and its spectral certificates. The proof combines local Neumann approximation, summable whole-cluster continuation, quantitative aperture bounds and crack-aware resolvent certification.

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Edition files

Available editions

  1. Edition 3 · 2026-11-01 · public
  2. Edition 2 · 2026-10-27 · public
  3. Edition 1 · 2026-10-22 · public

Research details

Subjects
Mathematics · Analysis and Spectral Theory
Keywords
Dirichlet Laplacian, dynamical localization, quasi-conical domain, spectral approximation, constructive geometry
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

Prescribed-weight operator bounds for entire bounded local source spaces on constructed quasi-conical Dirichlet towers, with arbitrary prescribed vanishing spectral filters and divergent energy rates, an all-energy obstruction, and effective rational finite-prefix geometry under supplied computable input promises.

Disclosures

OpenAI Codex assisted with research, proof development, programming, literature comparison and manuscript preparation. The author assumes responsibility for all content, including the mathematics and accompanying files.

Reproduction

Eighteen-page analytical manuscript and twelve-member package with four complete TeX sources, portable fixed-date build helper, exact finite arithmetic verifier and expected JSON, reader instructions, canonical citations and checksums. The documented Tectonic build reproduces the PDF exactly. Normal and optimized Python runs agree. Finite controls check aperture/mode/contour arithmetic; the PDE, continuum-tail and terminating-construction results rest on analytic proofs. Archive edition 2.

References and related resources

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