The Dirichlet-Neumann operator for a compact Riemannian manifold with smooth boundary has discrete spectrum, known as Steklov spectrum. This still holds if the boundary is only Lipschitz. However, as was first analyzed by Nazarov and Taskinen, essential spectrum appears when the boundary has cusp singularities.
I will consider the more general case of fibred cusp singularities (a simple example being the domain obtained by taking a ball B and removing from it a smaller ball touching B from the inside). I will give a precise description of the Schwartz kernel of the Dirichlet-Neumann operator near the singularity -- it lies in an adapted singular pseudodifferential calculus, the phi-calculus of Mazzeo and Melrose -- and a formula for the bottom of its essential spectrum. This is joint work with K. Fritzsch and E. Schrohe.
