Existence of bound states for layers built over hypersurfaces in Rn+1

The existence of discrete spectrum below the essential spectrum is deduced for the Dirichlet Laplacian on tubular neighborhoods (or layers) about hypersurfaces in Rn+1, with various geometric conditions imposed. This is a generalization of the results of Duclos, Exner, and Krejčiřík (2001) in the case of a surface in R3. The key to the generalization is the notion of parabolic manifolds. An interesting case in R3—that of the layer over a convex surface—is also investigated.