The Spectral Flow and the Maslov Index

exist and have no zero eigenvalue. A typical example for A(t) is the div-grad-curl operator on a 3-manifold twisted by a connection which depends on t. Atiyah et al proved that the Fredholm index of such an operator DA is equal to minus the “spectral flow” of the family {A(t)}t∈R. This spectral flow represents the net change in the number of negative eigenvalues of A(t) as t runs from −∞ to ∞. This “Fredholm index = spectral flow” theorem holds for rather general families {A(t)}t∈R of self-adjoint operators on Hilbert spaces. This is a folk theorem that has been used many times in the literature, but no adequate exposition has yet appeared. We give such an exposition here as well as several applications. More precisely, we shall prove the following theorem.