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Research Article · Preprint Mathematics

A Weyl Law for Rapidly Twisted Polar-Strip Tubes

Scott Schubr

  • 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 eigenvalue-counting asymptotic for a class of infinite-volume Dirichlet tubes confined by diverging twisting. The planar cross-section is a compact annular polar strip with positive C¹ angular width α(r) < 2π, and the twisting speed is c(1 + s²)^(p/2), where c, p > 0. We prove N(Λ) ∼ CΛ^(3/2 + 1/(2p)) and compute C explicitly in terms of ζ(1 + 1/p) and ∫ rα(r)^(1 + 1/p) dr. The proof retains the angular Dirichlet modes while treating the radial and longitudinal variables semiclassically. A uniform Dirichlet-box comparison obtained from quasiperiodic phase averaging and IMS localization controls the artificial longitudinal boundaries. A separate tail estimate and absolute-coefficient freezing justify the global limit. The result gives a concrete-class answer to the Weyl-asymptotics question for tubes with asymptotically diverging twisting.

Keywords Dirichlet Laplacian · twisted waveguide · Weyl law · infinite-volume domain · eigenvalue asymptotics

Scope of the work

The theorem assumes a compact annular polar strip away from the rotation axis, a positive C¹ angular width strictly below 2π, and twisting speed exactly c(1 + s²)^(p/2), with fixed c, p > 0. It proves the leading counting law and its eigenvalue and heat-trace consequences for this class. Arbitrary cross-sections, pinching widths, arbitrary twist profiles, a uniform limit in p, and a remainder estimate for the original operator are not asserted. This is a concrete-class partial answer to the general diverging-twist Weyl question.

Reproduction notes

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Author disclosures

AI tools assisted research, literature discovery, writing, and verification. Independent internal AI-assisted reviews are not external human refereeing or formal proof-assistant certification. The manuscript credits established confinement, bracketing, IMS localization, Fourier analysis, and discrete-mode asymptotic methods. External novelty and worldwide priority remain provisional.

References and related publications

  1. D. Barseghyan and P. Exner, Spectral estimates for the Dirichlet Laplacian on spiral-shaped regions, Journal of Spectral Theory 13 (2023), 243–261.
  2. D. Barseghyan and A. Khrabustovskyi, Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity, Operators and Matrices 13 (2019), 311–322.
  3. P. Briet, H. Kovařík, G. Raikov and E. Soccorsi, Eigenvalue asymptotics in a twisted waveguide, Communications in Partial Differential Equations 34 (2009), 818–836.
  4. V. Ivrii, 100 years of Weyl’s law, Bulletin of Mathematical Sciences 6 (2016), 379–452.
  5. D. Krejčiřík, Waveguides with asymptotically diverging twisting, Applied Mathematics Letters 46 (2015), 7–10.
  6. D. Krejčiřík, Spectral geometry: old questions and new answers, preprint (2026).

A stable scholarly record

Cite this edition

Scott Schubr. (2026). A Weyl Law for Rapidly Twisted Polar-Strip Tubes (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/a-weyl-law-for-rapidly-twisted-polar-strip-tubes/v/1

BibTeX
@techreport{luminary_a_weyl_law_for_rapidly_twisted_polar_strip_tubes_v1,
  title = {A Weyl Law for Rapidly Twisted Polar-Strip Tubes},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/a-weyl-law-for-rapidly-twisted-polar-strip-tubes/v/1}
}

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