Spectral flow and resonance index

Fredholm index and spectral flow

for all k. This shows that the sequence of functions is uniformly equicontinuous. Let now t ∈ R and assume that the sequence (|fk(t)|) is unbounded. Then it follows from the estimate that the sequence (min[t−1,t+1] |fk|) is unbounded and therefore also that (‖fk‖L2) is unbounded, contradiction. We conclude the the sequence ‖fk(t)‖ is bounded, for every t ∈ R. Let now T > 0. Then it follows by A...

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 ∞. ...

Dirac Index Classes and the Noncommutative Spectral Flow

We present a detailed proof of the existence-theorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997). We apply this result to various index-theoretic situations, extending to the noncommutative context results of Booss– Wojciechowski, Melrose–Piazza and Dai–Zhang. In particular, we prove a variational formula, in K * ðC r ðGÞÞ; for the ind...

The Maslov Index , the Spectral Flow , and Decompositions of Manifolds

On a Spectral Flow Formula for the Homological Index

Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators {A(t) | t ∈ R} that converges in norm to asymptotes A± at ±∞. Then under certain conditions [RoSa95] that include the assumption that the operators {D(t) = D + A(t), t ∈ R} all have discrete spectrum then the spectral flow along the path {D(t)} can be show...