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Research Article · Preprint Mathematics

Weyl Laws and Angular Stability for Rapidly Twisted 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 exact eigenvalue-counting asymptotics for infinite-volume Dirichlet tubes confined by rotation. Continuous regularly varying reference speeds of positive index require neither monotonicity nor derivative bounds; a second class includes smooth rapidly varying speeds such as exponential and double-exponential growth. The two axial ends may have different types. Finite-band cross-sections allow continuous angular endpoints, pinching widths, multiple components and finitely many topology transitions. The law remains unchanged under angle perturbations with finite limits, without control of their derivatives, and therefore permits infinitely many actual speed reversals and zeros. In the positive-index regime the coefficient is a nonlinear angular-width moment; in the rapid regime it depends only on area. We give full box and localization proofs, logarithmic trace thresholds, width-distribution recovery from a family of coefficients and a finite-data obstruction.

Keywords Dirichlet Laplacian · twisted tube · Weyl law · regular variation · angular stability · inverse spectral information

Scope of the work

The exact high-energy Dirichlet counting law applies to the stated finite-band cross-sections and positive-index regularly varying or specified smooth rapidly varying reference speeds. Angle perturbations must have finite limits; their derivatives need not be controlled. This substantially extends the author’s earlier polar-strip paper, which remains separately published and is credited. It is a concrete-class partial answer to the diverging-twist counting question. Index-zero slow variation, arbitrary fiber topology, effective remainders, Maxwell or Neumann operators, device performance, and full shape recovery from a single spectrum are not established.

Reproduction notes

The three-member package contains the complete twelve-page PDF, editable main.tex with all proofs and thirteen references, and reader instructions. Compile main.tex with Tectonic or run pdflatex twice. No external image, bibliography database, numerical data or research scripts are required. The paper gives full box and localization proofs; trace and moment consequences are identified separately from the main geometric theorem.

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Author disclosures

AI tools assisted manuscript preparation and internal analytic, attribution, and source-rebuild checks. The author is responsible for the final work. These checks do not constitute external human refereeing. Originality remains provisional. Classical regular-variation, horn asymptotics, spectral moments, and the previously published polar-strip box mechanism are credited; no general solution of the diverging-twist problem is claimed.

References and related publications

  1. N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge University Press, 1987.
  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. Y. Chitour, D. Prandi and L. Rizzi. Weyl’s law for singular Riemannian manifolds. Journal de Mathématiques Pures et Appliquées 181 (2024), 113–151.
  5. J. Fernández Bonder, J. P. Pinasco and A. M. Salort. Refined asymptotics for eigenvalues on domains of infinite measure. Journal of Mathematical Analysis and Applications 371 (2010), 41–56.
  6. V. Ivrii. 100 years of Weyl’s law. Bulletin of Mathematical Sciences 6 (2016), 379–452.
  7. D. Krejčiřík. Waveguides with asymptotically diverging twisting. Applied Mathematics Letters 46 (2015), 7–10.
  8. D. Krejčiřík. Spectral geometry: old questions and new answers. Preprint (2026), arXiv:2609.28602v1.
  9. D. Krejčiřík and R. Tiedra de Aldecoa. Ruled strips with asymptotically diverging twisting. Annales Henri Poincaré 19 (2018), 2069–2086.
  10. G. Raikov. Spectral asymptotics for waveguides with perturbed periodic twisting. Journal of Spectral Theory 6 (2016), 331–372.
  11. D. C. Suarez Bello. Spectral properties of the Dirichlet Laplacian in twisted and sheared waveguides. Letters in Mathematical Physics 116 (2026), article 119.
  12. S. Schubr. A Weyl Law for Rapidly Twisted Polar-Strip Tubes. Luminary Research Archive, edition 1 (2026).
  13. M. van den Berg and M. Lianantonakis. Asymptotics for the spectrum of the Dirichlet Laplacian on horn-shaped regions. Indiana University Mathematics Journal 50 (2001), 299–333.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Weyl Laws and Angular Stability for Rapidly Twisted Tubes (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/weyl-laws-angular-stability-rapidly-twisted-tubes/v/1

BibTeX
@techreport{luminary_weyl_laws_angular_stability_rapidly_twisted_tubes_v1,
  title = {Weyl Laws and Angular Stability for Rapidly Twisted Tubes},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/weyl-laws-angular-stability-rapidly-twisted-tubes/v/1}
}

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