Mayer-Vietoris formula for the determinant of a Laplace operator on an even-dimensional manifold

Abstract:

Let $\Delta$ be a Laplace operator acting on differential p-forms on an even-dimensional manifold M. Let $\Gamma$ be a submanifold of codimension 1. We show that if B is a Dirichlet boundary condition and R is a Dirichlet-Neumann operator on $\Gamma$, then ${\operatorname {Det}}(\Delta + \lambda ) = {\operatorname {Det}}(\Delta + \lambda ,B){\operatorname {Det}}(R + \lambda )$ and ${\operatorname {Det}^ \ast }\Delta = \frac {1}{{{{(\det A)}^2}}}{\operatorname {Det}}(\Delta ,B){\operatorname {Det}^ \ast }R$. This result was established in 1992 by Burghelea, Friedlander, and Rappeler for a 2-dimensional manifold with $p = 0$.