ON THE SPECTRAL FLOW OF FAMILIES OF DIRAC OPERATORS WITH CONSTANT SYMBOL

We consider families of generalized Dirac operators D(t) with constant principal symbol and constant essential spectrum such that the endpoi nts are gauge equivalent, i.e., D1 = WD0W. The spectral flow un any g ap in the essential spectrum we express as the Fredholm index of 1 + ( W - 1) P where P is the spectral projection on the interval [d, infini ty) with respect to D0 and d is in the gap. We reduce the computation of this index to the Atiyah-Singer index theorem for elliptic pseudodi fferential operators. We find an invariant of the Riemannian geometry for odd dimensional spin manifolds estimating the length of gaps in th e spectrum of the Dirac operator.