Location of the essential spectrum 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.

13 pages

References in corpus (2)

• On the spectrum of curved quantum waveguides
• Twisting versus bending in quantum waveguides

Cited by in corpus (11)

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• Spectral stability of Schroedinger operators with subordinated complex potentials
• The Hardy inequality and the heat equation with magnetic field in any dimension
• Dirac operator spectrum in tubes and layers with a zigzag type boundary
• Extreme localisation of eigenfunctions to one-dimensional high-contrast periodic problems with a defect
• Lifshits tails for randomly twisted quantum waveguides
• Spectral asymptotics for eigenvalues and resonances in the presence of a change of boundary conditions