A Hardy inequality in twisted waveguides

TL;DR

This paper demonstrates that twisting a straight, non-circular waveguide creates a Hardy inequality for the Laplacian, affecting the spectral stability under bending.

Contribution

It establishes a Hardy inequality for twisted waveguides and analyzes the spectral stability, showing the necessity of a critical bending strength for eigenvalue emergence.

Findings

Twisting induces a Hardy inequality for the Dirichlet Laplacian.

Small local bending does not produce eigenvalues below the essential spectrum.

A critical bending strength is required to generate discrete spectrum in rotated waveguides.

Abstract

We show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that any local bending, no matter how small, generates eigenvalues below the essential spectrum of the Laplacian in the tubes with arbitrary cross-sections rotated along a reference curve in an appropriate way. In the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a non-empty discrete spectrum.