Location of the essential spectrum in curved quantum layers
  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. 
David Krejcirik and Zhiqin Lu
math.DG math-ph math.FA math.MP math.SP