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 endpoints are gauge equivalent, i.e., D1 = W*D0W. The spectral flow un any gap 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, infinity) 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 pseudodifferential operators. We find an invariant of the Riemannian geometry for odd dimensional spin manifolds estimating the length of gaps in the spectrum of the Dirac operator.