Geometric Blind Modes in the Weyl Asymptotics of Twisted Tubes
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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.
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Research details
- Subjects
- Mathematics · Analysis and Spectral Theory
- Keywords
- twisted tube, Dirichlet Laplacian, Weyl law, log-periodicity, inverse spectral information, Mellin transform
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
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. 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. Both previously published twisted-tube papers remain separate and unchanged.
Disclosures
AI tools assisted mathematical exploration, drafting and independent internal checks, as disclosed in the manuscript. Classical transform-zero cancellation and log-periodic theory, earlier spectral work and the author’s two existing twisted-tube publications are credited.
Reproduction
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References and related resources
- N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation (1987)
- D. Barseghyan and A. Khrabustovskyi, Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity (2019)
- D. Krejčiřík, Waveguides with asymptotically diverging twisting (2015)
- P. Kevei, Regularly log-periodic functions and some applications (2020)
- M. L. Lapidus and H. Maier, The Riemann hypothesis and inverse spectral problems for fractal strings (1995)
- M. L. Lapidus, The Sound of Fractal Strings and the Riemann Hypothesis (2015)
- S. Schubr, A Weyl Law for Rapidly Twisted Polar-Strip Tubes, edition 1 (2026)
- S. Schubr, Weyl Laws and Angular Stability for Rapidly Twisted Tubes, edition 1 (2026)
- M. van den Berg and M. Lianantonakis, Asymptotics for the spectrum of the Dirichlet Laplacian on horn-shaped regions (2001)