ON THE DETERMINANT OF ELLIPTIC DIFFERENTIAL AND FINITE-DIFFERENCE OPERATORS IN VECTOR-BUNDLES OVER S1

For an elliptic differential operator A over S1, A = [GRAPHICS] A(k)(x)D(k), with A(k)(x) in END (C(r)) and theta as a principal angle, the zeta-regularized determinant Det-theta 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.