A Weyl Law for Rapidly Twisted Polar-Strip Tubes
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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.
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Research details
- Subjects
- Mathematics · Analysis and Spectral Theory
- Keywords
- Dirichlet Laplacian, twisted waveguide, Weyl law, infinite-volume domain, eigenvalue asymptotics
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
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. This is a concrete-class partial answer to the general diverging-twist Weyl question.
Disclosures
AI tools assisted research, literature discovery, writing, and verification. The manuscript credits established confinement, bracketing, IMS localization, Fourier analysis, and discrete-mode asymptotic methods.
Reproduction
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References and related resources
- D. Barseghyan and P. Exner, Spectral estimates for the Dirichlet Laplacian on spiral-shaped regions, Journal of Spectral Theory 13 (2023), 243–261.
- D. Barseghyan and A. Khrabustovskyi, Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity, Operators and Matrices 13 (2019), 311–322.
- P. Briet, H. Kovařík, G. Raikov and E. Soccorsi, Eigenvalue asymptotics in a twisted waveguide, Communications in Partial Differential Equations 34 (2009), 818–836.
- V. Ivrii, 100 years of Weyl’s law, Bulletin of Mathematical Sciences 6 (2016), 379–452.
- D. Krejčiřík, Waveguides with asymptotically diverging twisting, Applied Mathematics Letters 46 (2015), 7–10.
- D. Krejčiřík, Spectral geometry: old questions and new answers, preprint (2026).