The nature of the essential spectrum in curved quantum waveguides

TL;DR

This paper investigates the essential spectrum of the Dirichlet Laplacian in curved quantum waveguides, establishing conditions for the absence of singular continuous spectrum and analyzing embedded eigenvalues using Mourre theory.

Contribution

It provides new insights into the spectral properties of Laplacians in curved waveguides, extending Mourre theory to these geometries and analyzing embedded eigenvalues.

Findings

Absence of singular continuous spectrum under decay conditions.

Characterization of embedded eigenvalues.

Spectral analysis of Schrödinger operators in straight tubes.

Abstract

We study the nature of the essential spectrum of the Dirichlet Laplacian in tubes about infinite curves embedded in Euclidean spaces. Under suitable assumptions about the decay of curvatures at infinity, we prove the absence of singular continuous spectrum and state properties of possible embedded eigenvalues. The argument is based on Mourre conjugate operator method developed for acoustic multistratified domains by Benbernou and Dermenjian et al. As a technical preliminary, we carry out a spectral analysis for Schrodinger-type operators in straight Dirichlet tubes. We also apply the result to the strips embedded in abstract surfaces.