A sharp flatness transition for critical heat traces in twisted tubes

Abstract

We study the Dirichlet Laplacian in a three-dimensional tube with fixed annular-strip cross-section and twisting speed proportional to the square root of a logarithm. Its eigenvalue count has a leading exponential rate, which determines a critical heat-trace time but does not determine finiteness at that time. We classify the critical trace when the angular confinement potential has an infinitely flat isolated minimum of the form a exp(-A/|r-r*|^q). The trace is finite for q<2 and infinite for q>2. At q=2 it is finite exactly when A<Kπ²/6, where K is the critical time; equality gives divergence. Finite-order isolated width maxima also give a finite critical trace, whereas a width plateau gives divergence. The proof retains the original coupled Dirichlet form. A constant-twist heat comparison, using diamagnetic domination, preserves the helical-mode multiplicity. A family of exactly orthogonal helical trial functions supplies the matching lower bound, including the critical parameter boundary.