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.

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.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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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: | https://doi.org/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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