Location of the essential spectrum in curved quantum layers

TL;DR

This paper investigates the essential spectrum of the Dirichlet Laplacian in curved quantum layers, showing it matches that of a planar tube under certain geometric conditions, relevant for nanostructure physics.

Contribution

It establishes conditions under which the essential spectrum in curved quantum layers aligns with that of a flat, planar tube, extending spectral analysis to curved geometries.

Findings

Essential spectrum coincides with that of a planar tube under specific geometric decay conditions.

Results apply to low-dimensional nanostructures with natural physical assumptions.

Provides a geometric criterion for spectral localization in curved quantum layers.

Abstract

We consider the Dirichlet Laplacian in tubular neighbourhoods of complete non-compact Riemannian manifolds immersed in the Euclidean space. We show that the essential spectrum coincides with the spectrum of a planar tube provided that the second fundamental form of the manifold vanishes at infinity and the transport of the cross-section along the manifold is asymptotically parallel. For low dimensions and codimensions, the result applies to the location of propagating states in nanostructures under physically natural conditions.