Spectra of the Dirichlet Laplacian in 3-dimensional polyhedral layers

The structure of the spectrum of the three-dimensional Dirichlet Laplacian in a 3D polyhedral layer of fixed width is studied. It turns out that the essential spectrum is determined by the smallest dihedral angle that forms the boundary of the layer while the discrete spectrum is always finite. An example of a layer with empty discrete spectrum is constructed. The spectrum is proved to be nonempty in regular polyhedral layer.