Relative zeta-determinants of Laplacians on a manifold with cylindrical ends

초록

On a compact manifold with boundary a Laplacian subject to some elliptic boundary condition has discrete spectrum and the zeta function for the Laplacian is well-defined from this spectrum. However, on a non-compact manifold the spectrum of a Laplacian may have a continuous spectrum and hence the zeta function cannot be defined directly. To avoid this difficulty some mathematicians suggested the concept of relative zeta-determinant. In this note we discuss briefly the concepts of zeta- and relative zeta-determinants of Laplacians and give some results for the relative zeta-determinants of Laplacians on a manifold with cylindrical ends.