On the Fredholm Lagrangian Grassmannian, Spectral Flow and ODEs in Hilbert Spaces

TL;DR

This paper extends spectral flow formulas and the Maslov index to infinite-dimensional symplectic Hilbert spaces, relating spectra of Hamiltonian systems with homoclinic boundary conditions to stable-unstable space intersections.

Contribution

It generalizes finite-dimensional spectral flow and Maslov index results to infinite-dimensional Hilbert spaces, introducing new index bundle concepts for unbounded operators.

Findings

Extended spectral flow formulas to Hilbert spaces.

Generalized Maslov index to infinite dimensions.

Connected spectra of Hamiltonian systems to stable-unstable space intersections.

Abstract

We consider homoclinic solutions for Hamiltonian systems in symplectic Hilbert spaces and generalise spectral flow formulas that were proved by Pejsachowicz and the author in finite dimensions some years ago. Roughly speaking, our main theorem relates the spectra of infinite dimensional Hamiltonian systems under homoclinic boundary conditions to intersections of their stable and unstable spaces. Our proof has some interest in its own. Firstly, we extend a celebrated theorem by Cappell, Lee and Miller about the classical Maslov index in $\mathbb{R}^{2n}$ to symplectic Hilbert spaces. Secondly, we generalise the classical index bundle for families of Fredholm operators of Atiyah and J\"anich to unbounded operators for applying it to Hamiltonian systems under varying boundary conditions. Finally, we substantially make use of striking results by Abbondandolo and Majer to study Fredholm properties of infinite dimensional Hamiltonian systems.