Eigenvalue Asymptotics in a Twisted Waveguide

TL;DR

This paper analyzes the spectral properties of a twisted quantum waveguide, showing that slow decay of the twist leads to infinitely many bound states and deriving their asymptotic behavior.

Contribution

It provides new insights into the eigenvalue asymptotics of the Dirichlet Laplacian in twisted waveguides with slowly decaying twist.

Findings

Existence of infinitely many discrete eigenvalues below the essential spectrum.

Asymptotic formula for the eigenvalue sequence.

Dependence of eigenvalues on the decay rate of the twist derivative.

Abstract

We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.