arXiv subjects
Publications and source records attributed to Krzysztof P. Wojciechowski.
We study an example of an index problem for a Dirac-like operator subject to Atiyah-Patodi-Singer boundary conditions on a noncommutative manifold with boundary, namely the quantum unit disk.
We discuss the decomposition of the zeta-determinant of the square of the Dirac operator into contributions coming from the different parts of the manifold. The easy case was worked in the previous paper of authors. Due to the assumptions made on the operators in the previous paper, we were able to avoid the presence of the small eigenvalues which provide the large time contibution to the determinant. In the present work we analyze the general case. We offer a detailed analysis of the contribution made by the small eigenvalues.
we discuss the decomposition of the zeta-determinant of the square of the Dirac operator into the contributions coming from the different parts of the manifold in the case of an invertible tangential operator.
We prove an adiabatic decomposition formula of the zeta-determinant of the Laplace type operator with respect to Dirichlet boundary condition. We allow the non-invertible tangential operator. As a result, our adiabatic decomposition formula involves the scattering matrix over the manifold with cylindrical end. We also describe the adiabatic limit of the zeta-determinant of the Dirichlet to Neumann operator, which plays the essential role of Burghelea-Friedlander-Kappeler's Meyer-Vietoris type formula of the zeta determinant.
We review the work of the authors and their collaborators on the decomposition of the zeta-determinant of the Dirac operator into the contribution coming from different parts of a manifold.