Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity

TL;DR

This paper investigates the spectral properties of the Dirichlet Laplacian on unbounded twisted tubes with rapidly increasing twisting velocity, establishing a Berezin type upper bound for eigenvalue moments.

Contribution

It provides a new spectral estimate (Berezin type upper bound) for the eigenvalues of the Laplacian in twisted tubes with exploding twisting velocity, extending previous results.

Findings

Spectrum is purely discrete under certain conditions.

Established a Berezin type upper bound for eigenvalue moments.

Extended spectral analysis to tubes with unbounded twisting velocity.

Abstract

We study the spectrum of the Dirichlet Laplacian on an unbounded twisted tube with twisting velocity exploding to infinity. If the tube cross section does not intersect the axis of rotation, then its spectrum is purely discrete under some additional conditions on the twisting velocity (D.Krejcirik, 2015). In the current work we prove a Berezin type upper bound for the eigenvalue moments.