Regularized determinants of Laplace type operators, analytic surgery and   relative determinants
  We study the asymptotic behaviour of regularized determinants of certain Laplace type operators with respect to singular deformations of the underlying manifold which are obtained by stretching a tubular neighborhood of an embedded separating hypersurface to a cylinder of infinite length. Using the asymptotic expansions, we derive adiabatic splitting formulas for regularized determinants of Laplace type operators. 
Joern Mueller, Werner Mueller