### Abstract

For an elliptic differential operator A over S^{1}, {Mathematical expression}, with A_{k}(x) in END(ℂ^{r}) and θ as a principal angle, the ζ-regularized determinant Det_{θ}A is computed in terms of the monodromy map P_{A}, associated to A and some invariant expressed in terms of A_{n} and A_{n-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 self adjoint and show that the determinant of A is the limit of a sequence of computable expressions involving determinants of difference approximation of A.

Original language | English (US) |
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Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Communications in Mathematical Physics |

Volume | 138 |

Issue number | 1 |

DOIs | |

State | Published - May 1 1991 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Burghelea, D., Friedlander, L., & Kappeler, T. (1991). On the determination of elliptic differential and finite difference operators in vector bundles over S

^{1}.*Communications in Mathematical Physics*,*138*(1), 1-18. https://doi.org/10.1007/BF02099666