Geometric Blind Modes in the Weyl Asymptotics of Twisted Tubes

Overview

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.

Keywords twisted tube · Dirichlet Laplacian · Weyl law · log-periodicity · inverse spectral information · Mellin transform