Gluing Formula for Casimir Energies ^†

Let M₁ and M₂ be two Riemannian manifolds each of which has the boundary N. Consider the Laplacian on M₁ and M₂ augmented with Dirichlet boundary conditions on N. A natural question to ask is whether there is any relation between spectral properties of the Laplacian on M₁, M₂, and the Laplacian on the manifold M (without boundary) obtained by gluing together M₁ and M₂, namely M = M₁ ∪_N M₂. A partial answer is given by the Burghelea-Friedlander-Kappeler-gluing formula for zeta-determinants. This formula contains an (in general) unknown polynomial which is completely determined by some data on a collar neighborhood of the hypersurface N. In this talk, I present results for the polynomial in terms of suitable geometric tensors on N. Choosing M₁, M₂ and M as appropriate, results in a gluing formula for Casimir energies.