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Weyl Laws and Angular Stability for Rapidly Twisted Tubes

Scott Schubr

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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.

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Edition files

Available editions

  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-22 · public

Research details

Subjects
Mathematics · Analysis and Spectral Theory
Keywords
Dirichlet Laplacian, twisted tube, Weyl law, regular variation, angular stability, inverse spectral information
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

Exact two-ended Weyl laws for the specified regularly or rapidly varying reference speeds and finite-band cross-sections, stable under convergent angle perturbations without derivative control; leading coefficients determine a width distribution across a family of operators.

Disclosures

OpenAI Codex assisted with literature comparison, proof review and manuscript preparation. The author takes responsibility for the content, accuracy and attribution of the work.

Reproduction

The 12-page searchable manuscript is accompanied by complete editable sources, reader instructions, citation metadata and integrity checksums. The manuscript supplies the analytic arguments; accompanying computational materials, where included, check their specified finite certificates.

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