The Maslov Index and the Spectral Flow - revisited

TL;DR

This paper provides an elementary proof of a theorem linking the Maslov index to spectral flow, emphasizing operator continuity and extending results to linear Hamiltonian systems.

Contribution

It offers a simplified proof of a key theorem relating Maslov index and spectral flow, and generalizes spectral flow formulas for Hamiltonian systems.

Findings

Elementary proof of Cappell, Lee, and Miller's theorem

Continuity analysis of operator paths using gap-metric

Generalized spectral flow formula for Hamiltonian systems

Abstract

We give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of selfadjoint first-order operators. We particularly pay attention to the continuity of the latter path of operators, where we consider the gap-metric on the set of all closed operators on a Hilbert space. Finally, we obtain from Cappell, Lee and Miller's theorem a spectral flow formula for linear Hamiltonian systems which generalises a recent result of Hu and Portaluri.