The Maslov Index: a Functional Analytical Definition and the Spectral Flow Formula

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 ∞. ...

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...

An Extension of a Theorem of Nicolaescu on Spectral Flow and the Maslov Index

In this paper we extend a theorem of Nicolaescu on spectral flow and the Maslov index. We do this by studying the manifold of Lagrangian subspaces of a symplectic Hilbert space that are Fredholm with respect to a given Lagrangian L0. In particular, we consider the neighborhoods in this manifold of Lagrangians which intersect L0 nontrivially.

Computation of the Maslov Index and the Spectral Flow via Partial Signatures

Given a smooth Lagrangian path, both in the finite and in the infinite dimensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the Maslov cycle. For real-analytic paths, we give a formula for the computation of the Maslov index using the partial signatures; a similar formula holds for the spectral flow of real-analytic paths of F...