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)=Λ^2/(6π)+o(Λ^2) 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(Λ^(7/4)) 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
An explicit periodic leading Dirichlet count, a geometry-dependent Fourier recovery criterion and an infinite-dimensional blind-mode example for logarithmically periodic inverse twisting speeds; constant-step geometries also admit a quantitative remainder and a detectable lower-homogeneity profile.
Disclosures
AI tools assisted mathematical exploration, drafting and independent internal checks, as disclosed in the manuscript. These checks are not external human refereeing or formal proof verification. Originality remains provisional. Classical transform-zero cancellation and log-periodic theory, earlier spectral work and the author’s two existing twisted-tube publications are credited. The pure-model lower-order question remains open.
Reproduction
The paper has 13 pages. The editable-source archive contains 6 files. Compile main.tex; the analytic proofs and bibliography are self-contained.
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)