On a Comparison Principle and the Uniqueness of Spectral Flow

TL;DR

This paper investigates the properties and uniqueness of the spectral flow, providing new formulas and approaches that extend its applicability, and explores its relation to the Maslov index in symplectic Hilbert spaces.

Contribution

It introduces a new comparison theorem for spectral flow, removes the invertibility assumption for its uniqueness, and links it to the Maslov index.

Findings

Established a simple formula for homotopy invariance of spectral flow.

Proved a new approach to spectral flow's uniqueness without endpoint invertibility.

Connected spectral flow with the Maslov index in symplectic Hilbert spaces.

Abstract

The spectral flow is a well-known quantity in spectral theory that measures the variation of spectra about $0$ along paths of selfadjoint Fredholm operators. The aim of this work is twofold. Firstly, we consider homotopy invariance properties of the spectral flow and establish a simple formula which comprises its classical homotopy invariance and yields a comparison theorem for the spectral flow under compact perturbations. We apply our result to the existence of non-trivial solutions of boundary value problems of Hamiltonian systems. Secondly, the spectral flow was axiomatically characterised by Lesch, and by Ciriza, Fitzpatrick and Pejsachowicz under the assumption that the endpoints of the paths of selfadjoint Fredholm operators are invertible. We propose a different approach to the uniqueness of spectral flow which lifts this additional assumption. As application of the latter result, we discuss the relation between the spectral flow and the Maslov index in symplectic Hilbert spaces.