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

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.