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Adiabatic decomposition of the zeta-determinant and Scattering theory | | Jinsung Park
; Krzysztof P. Wojciechowski
; | | Date: |
5 Nov 2001 | | Subject: | Differential Geometry; Spectral Theory MSC-class: 58J52, 58J32, 81Q70 | math.DG math.SP | | Abstract: | 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. The idea is based on the method used by Werner M"uller in his paper where he studied the eta-invariant on a manifold with cylindrical end. The analysis of the zeta-determinant, however, is more difficult. The difference is illustrated nicely in the proof of the Lesch-Wojciechowski formula for the ratio of the zeta-determinants of the Dirac operator subject to the two different boundary conditions provided in the final sections of the paper. | | Source: | arXiv, math.DG/0111046 | | Services: | Forum | Review | PDF | Favorites | |
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