Adiabatic decomposition of the ζ-determinant and Dirichlet to Neumann operator

We discuss the adiabatic decomposition formula of the ζ-determinant of a Laplace type operator on a closed manifold. We also analyze the adiabatic behavior of the ζ-determinant of a Dirichlet to Neumann operator. This analysis makes it possible to compare the adiabatic decomposition formula with the Mayer-Vietoris type formula for the ζ-determinant proved by Burghelea et al. As a byproduct of this comparison, we obtain the exact value of the local constant which appears in their formula for the case of Dirichlet boundary condition.