Yoonweon Lee

Burghelea-Friedlander-Kappeler's gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion

The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under an assumption that the product structure is given near the boundary. As applications of this result, we prove the adiabatic decomposition theorems of the zeta-determinant of a Laplacian with respect to the Dirichlet and Neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions.

ADIABATIC DECOMPOSITION OF THE -DETERMINANT OF THE DIRAC LAPLACIAN. I. THE CASE OF AN INVERTIBLE TANGENTIAL OPERATOR

ABSTRACT We discuss the decomposition of the ζ-determinant of the square of the Dirac operator into the contributions coming from the different parts of the manifold. The result was announced in the Note Ref. [16]. The proof sketched in the Note was based on results of Brüning and Lesch (see Ref. [4]). In the meantime we have found another proof, more direct and elementary, and closer to the spirit of the original papers which initiated the study of the adiabatic decomposition of the spectral invariants (see Refs. [7] and [21]). We discuss this proof in detail. We study the general case (non-invertible tangential operator) in forthcoming work (see Refs. [17] and [18]). In the Appendix we present the computation of the cylinder contribution to the ζ-function of the Dirac Laplacian on a manifold with boundary, which we need in the main body of the paper. This computation is also used to show the vanishing result for the ζ-function on a manifold with boundary.

Mayer-Vietoris formula for determinants of elliptic operators of Laplace-Beltrami type (after Burghelea, Friedlander and Kappeler)

Abstract The purpose of this note is to provide a short cut presentation of a Mayer-Vietoris formula due to Burghelea-Friedlander-Kappeler for the regularized determinant in the case of elliptic operators of Laplace-Beltrami type in the form typically needed in applications to torsion.

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

Let A be a Laplace operator acting on differential p-forms on an even-dimensional manifold M. Let F be a submanifold of codimension 1. We show that if B is a Dirichlet boundary condition and R is a DirichletNeumann operator on r, then Det(A + A) = Det(A + A, B) Det(R + A) and Det* a = Det(A, B) Det* R. This result was established in 1992 by Burghelea, Friedlander, and Kappeler for a 2-dimensional manifold with p = 0.

THE REFINED ANALYTIC TORSION AND A WELL-POSED BOUNDARY CONDITION FOR THE ODD SIGNATURE OPERATOR

In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary under some assumption. For this purpose we introduce two boundary conditions which are complementary to each other and well-posed for the odd signature operator

The Burghelea-Friedlander-Kappeler–gluing formula for zeta-determinants on a warped product manifold and a product manifold

\mathcal{B}

THE BFK-GLUING FORMULA FOR ZETA-DETERMINANTS AND THE VALUE OF RELATIVE ZETA FUNCTIONS AT ZERO

in the sense of Seeley. We then show that the zeta-determinants of

Asymptotic expansions of the zeta-determinants of Dirac Laplacians on a compact manifold with boundary

\mathcal{B}^2

Gluing Formula for Casimir Energies

and eta-invariants of

A coefficient of an asymptotic expansion of logarithms of determinants for classical elliptic pseudodifferential operators with parameters

\mathcal{B}