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

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

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