Adiabatic Decomposition of the Ζ-determinant and Dirichlet to Neumann Operator

Abstract. This paper is companion to our earlier work [8] (see also announcement [7]). Let M be a closed manifold and Y be an embedded hypersurface, such that there is a decomposition of M = M1 ∪M2 into two manifolds with boundary M1 and M2 , with M1 ∩M2 = Y . In [8] we proved the decomposition formula for detζ∆ the ζ-determinant of a Dirac Laplacian ∆ on M . The contributions coming from M1 and M2 were described in terms of the ζ-determinants of the boundary problems of Atiyah-Partodi-Singer type and the scattering matrices defined by the Dirac operator . We used adiabatic process to split non-local ζ-determinant onto corresponding pieces. In this papper we discuss a similar result for Laplace type operators and Dirichlet boundary conditions. We closely follow the proof described in [8]. The significant difference is in the analysis of the relation between the small eigenvalues and the scattering matrices. We are also able to analyze the adiabatic behaviour of Dirichlet to Neumann operator, which allows us to make a connection with Mayer-Viertoris type formula for the ζ-determinant obtained by Burghelea, Friedlander and Kappeler. As a byproduct, we evaluate the exact value of the local constant which appears in their formula for the Dirichlet boundary condition.