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On the determinant of elliptic differential and finite difference operators in vector bundles over S¹


Burghelea, D; Friedlander, L; Kappeler, T (1991). On the determinant of elliptic differential and finite difference operators in vector bundles over S¹. Communications in Mathematical Physics, 138(1):1-18.

Abstract

In this paper we study the determinant of elliptic differential operators on a complex vector bundle E→pM of rank N over a compact oriented connected manifold M of dimension 1, as well as the determinants of its finite difference approximations. For an elliptic differential operator A over S1, A=∑nk=0Ak(x)Dk, with Ak(x) in END(Cr) and θ as a principal angle, the ζ-regularized determinant DetθA is computed in terms of the monodromy map PA associated to A and some invariant expressed in terms of An and An−1. A similar formula holds for finite difference operators. A number of applications and implications are given. In particular, we present a formula for the signature of A when A is selfadjoint and show that the determinant of A is the limit of a sequence of computable expressions involving determinants of difference approximations of A.

In this paper we study the determinant of elliptic differential operators on a complex vector bundle E→pM of rank N over a compact oriented connected manifold M of dimension 1, as well as the determinants of its finite difference approximations. For an elliptic differential operator A over S1, A=∑nk=0Ak(x)Dk, with Ak(x) in END(Cr) and θ as a principal angle, the ζ-regularized determinant DetθA is computed in terms of the monodromy map PA associated to A and some invariant expressed in terms of An and An−1. A similar formula holds for finite difference operators. A number of applications and implications are given. In particular, we present a formula for the signature of A when A is selfadjoint and show that the determinant of A is the limit of a sequence of computable expressions involving determinants of difference approximations of A.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1991
Deposited On:29 Nov 2010 16:29
Last Modified:05 Apr 2016 13:28
Publisher:Springer
ISSN:0010-3616
Additional Information:Erratum 1992 http://www.zora.uzh.ch/22727/
Publisher DOI:10.1007/BF02099666
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0734.58043
http://projecteuclid.org/euclid.cmp/1104202844
http://www.zora.uzh.ch/22727/

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