Existence of discrete eigenvalues for the Dirichlet Laplacian in a two-dimensional twisted strip

TL;DR

This paper investigates how twisting a two-dimensional strip affects the spectrum of the Dirichlet Laplacian, demonstrating that local twisting can generate discrete eigenvalues and analyzing their asymptotic behavior under specific conditions.

Contribution

It establishes the existence of discrete eigenvalues caused by local twisting and characterizes their asymptotic behavior when the twist grows at infinity and the strip narrows.

Findings

Local twisting induces discrete eigenvalues.

Asymptotic behavior of eigenvalues is characterized.

Twisting effects can be significant even at infinity.

Abstract

We study the spectrum of the Dirichlet Laplacian operator in a two-dimensional twisted strip embedded in $\mathbb R^d$ with $d \geq 2$. It is shown that a local twisting perturbation can create discrete eigenvalues for the operator. In particular, we also study the case where the twisted effect "grows" at infinity while the width of the strip goes to zero. In this situation, we find an asymptotic behavior for the eigenvalues.