Spectrum of the Schr\"odinger operator in a perturbed periodically twisted tube

TL;DR

This paper investigates how local modifications to the twisting of a straight, screw-shaped tube with a non-circular cross section can induce discrete eigenvalues in the spectrum of the Dirichlet Laplacian, revealing spectral changes due to geometric perturbations.

Contribution

It demonstrates that slowing down the twisting in a mean sense creates discrete spectrum in a twisted tube, advancing understanding of spectral effects of geometric perturbations.

Findings

Local slowing down of twisting induces discrete spectrum

Spectral changes depend on mean perturbation of twist

Non-empty discrete spectrum arises from geometric modifications

Abstract

We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is shown that a local perturbation which consists of "slowing down" the twisting in the mean gives rise to a non-empty discrete spectrum.