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Research Article · Preprint Mathematics

Geometric Blind Modes in the Weyl Asymptotics of Twisted Tubes

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Analysis and Spectral Theory
Primary subject
Mathematics
Subject subfield
Analysis and Spectral Theory
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

Abstract

We determine the leading Dirichlet eigenvalue count for rapidly twisted Euclidean tubes whose inverse twisting speed is a power times a smooth positive logarithmically periodic function. The actual operator count is an explicit periodic profile, without a regular-variation hypothesis. The Fourier multiplier factors into a nonvanishing universal factor and a complex angular-width moment. This gives a necessary and sufficient criterion for recovery of the inverse-speed modulation from the complete leading profile. A fixed connected two-width cross-section hides every odd logarithmic harmonic. In particular, smooth strictly increasing non-regularly-varying speeds on this same geometry have the pure-power law N(Λ) = Λ²/(6π) + o(Λ²) for two identical ends. The ambiguity is infinite dimensional, whereas visible modes admit conditional Sobolev recovery. For constant-step sections and inverse speeds with derivative bounded above and away from zero, we also prove an O(Λ⁷⁄⁴) counting remainder. A mixed-homogeneity inverse speed then has a genuine second periodic spectral profile: every mode of its lower-order modulation is visible on the example geometry. Whether lower terms reveal the original pure blind modulation remains open, as does equality of complete spectra.

Keywords twisted tube · Dirichlet Laplacian · Weyl law · log-periodicity · inverse spectral information · Mellin transform

Scope of the work

The leading-profile theorem applies to the stated rapidly twisted Euclidean tubes and logarithmically periodic inverse speeds on admissible finite-band cross-sections. Blindness concerns the complete leading counting profile on a fixed known geometry, not equality of full spectra. The quantitative O(Λ⁷⁄⁴) remainder has narrower constant-step hypotheses. Actual second-profile recovery applies to the mixed-homogeneity class with 1/2 < δ < 1; it does not prove lower-order audibility of the original pure critical blind model. No exact isospectrality, optimal remainder, finite-spectrum stable inversion or Riemann-hypothesis result is asserted. Both previously published twisted-tube papers remain separate and unchanged.

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Author disclosures

AI tools assisted mathematical exploration, drafting and independent internal checks, as disclosed in the manuscript. These checks are not external human refereeing or formal proof verification. Originality remains provisional. Classical transform-zero cancellation and log-periodic theory, earlier spectral work and the author’s two existing twisted-tube publications are credited. The pure-model lower-order question remains open.

References and related publications

  1. N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation (1987)
  2. D. Barseghyan and A. Khrabustovskyi, Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity (2019)
  3. D. Krejčiřík, Waveguides with asymptotically diverging twisting (2015)
  4. P. Kevei, Regularly log-periodic functions and some applications (2020)
  5. M. L. Lapidus and H. Maier, The Riemann hypothesis and inverse spectral problems for fractal strings (1995)
  6. M. L. Lapidus, The Sound of Fractal Strings and the Riemann Hypothesis (2015)
  7. S. Schubr, A Weyl Law for Rapidly Twisted Polar-Strip Tubes, edition 1 (2026)
  8. S. Schubr, Weyl Laws and Angular Stability for Rapidly Twisted Tubes, edition 1 (2026)
  9. M. van den Berg and M. Lianantonakis, Asymptotics for the spectrum of the Dirichlet Laplacian on horn-shaped regions (2001)

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Geometric Blind Modes in the Weyl Asymptotics of Twisted Tubes (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/geometric-blind-modes-weyl-asymptotics-twisted-tubes/v/1

BibTeX
@techreport{luminary_geometric_blind_modes_weyl_asymptotics_twisted_tubes_v1,
  title = {Geometric Blind Modes in the Weyl Asymptotics of Twisted Tubes},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/geometric-blind-modes-weyl-asymptotics-twisted-tubes/v/1}
}

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