Ruled strips with asymptotically diverging twisting

TL;DR

This paper investigates the spectral properties of a 2D Dirichlet Laplacian in a strip with diverging twisting, revealing a dimensional transition at infinity and the existence of discrete eigenvalues in certain geometries.

Contribution

It introduces a model with diverging twisting in a strip, demonstrating a dimensional transition and spectral characteristics, including discrete eigenvalues for specific cross-sections.

Findings

Essential spectrum determined by an asymptotic 3D tube.

Existence of discrete eigenvalues below the essential spectrum for disk cross-sections.

Model exhibits a 'raise of dimension' at infinity.

Abstract

We consider the Dirichlet Laplacian in a two-dimensional strip composed of segments translated along a straight line with respect to a rotation angle with velocity diverging at infinity. We show that this model exhibits a "raise of dimension" at infinity leading to an essential spectrum determined by an asymptotic three-dimensional tube of annular cross-section. If the cross-section of the asymptotic tube is a disk, we also prove the existence of discrete eigenvalues below the essential spectrum.