##
**On the determinant of elliptic differential and difference finite operators in vector bundles over \(S^ 1\).**
*(English)*
Zbl 0734.58043

The authors study the determinant (\(\zeta\)-regularized determinant) of elliptic differential operators on \(S^ 1\). A formula for it is given in terms of the monodromy map, the determinant of the monodromy matrix and an invariant relating with the splitting of the vector bundle on which the differential operator is defined. Also a similar formula is proved for finite difference operators and an approximation formula of the determinant of the differential operator is given by that of the finite difference approximation of the differential operator. Some applications are given, in particular, a formula is proved of the \(\eta\)-invariant when the operator is selfadjoint. The authors note that the method presented here is suited for the generalization to the case of pseudodifferential operators.

Reviewer: K.Furutani (Chiba-ken)

### MSC:

58J52 | Determinants and determinant bundles, analytic torsion |

### Keywords:

determinant of elliptic operators; \(\zeta\)-function; finite difference operator; monodromy map
PDF
BibTeX
XML
Cite

\textit{D. Burghelea} et al., Commun. Math. Phys. 138, No. 1, 1--18 (1991; Zbl 0734.58043)

Full Text:
DOI

### References:

[1] | [A] Arnold, V.: Ordinary Differential equations. Cambridge: MIT Press 1973 · Zbl 0296.34001 |

[2] | [APS] Atiyah, M., Patodi, V. K., Singer, I. M.: Spectral asymmetry and Riemannian geometry. Bull. London Math. Soc.55, 229–234 (1973) · Zbl 0268.58010 |

[3] | [DD] Dreyfus, T., Dym, H.: Procuct formulas for the eigenvalues of a class of boundary value problems. Duke Math. J.45, 15–37 (1978) · Zbl 0387.34021 |

[4] | [F] Forman, R.: Functional determinants and geometry. Invent. Math.88, 447–493 (1987) · Zbl 0602.58044 |

[5] | [Fr] Friedlander, L.: Determinants of Elliptic operators, Thesis MIT, 1989 |

[6] | [G] Guillemin, V.: A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues. Adv. Math.55, 131–160 (1985) · Zbl 0559.58025 |

[7] | [Gi] Gilkey, P.: The {\(\eta\)}-invariant and secondary characteristic classes for locally flat bundles. Teubner Texte zur Mathematik Band 17, pp. 49 |

[8] | [H] Hille, E.: Lectures on ordinary differential equations. Reading, MA: Addison-Wesley 1969 · Zbl 0179.40301 |

[9] | [K] Kato, T.: Perturbation theory for liner operators, 2nd ed. Berlin, Heidelberg, New York: Springer (1976) |

[10] | [N] Naimark, M. A.: Linear Differential Operators, Part I. Frederick Ungar 1967 · Zbl 0219.34001 |

[11] | [Se] Seeley, R.: Complex powers of elliptic operators. Proc. Symp. on Singular Integrals, AMS10, 288–307 (1967) |

[12] | [Si] Simon, B.: Notes on Infinite Determinants of Hilbert Space Operators. Adv. Math.24, 244–273 (1977) · Zbl 0353.47008 |

[13] | [Sh] Shubin, M. A.: Pseudodifferential operators and spectral theory. Berlin, Heidelberg, New York: Springer 1987 · Zbl 0616.47040 |

[14] | [T] Titchmarsh, E.: The Riemann zeta-function. Cambridge: Cambridge Press 1986 · Zbl 0601.10026 |

[15] | [W] Wodzicki, M.: Noncommutative residue inK-Theory, Arithmetic and Geometry. Manin, Yu. I. (ed.). Lecture Notes in Mathematics, vol.1289. Berlin, Heidelberg, New York: Springer 1987 · Zbl 0649.58033 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.