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

For an elliptic differential operator A over S¹, {Mathematical expression}, with A_k(x) in END(ℂ^r) and θ as a principal angle, the ζ-regularized determinant Det_θA 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.