Floer homology for Hamiltonian PDEs: Fredholm theory

TL;DR

This paper develops a Floer homology framework for infinite-dimensional Hamiltonian PDEs by establishing Fredholm properties of the linearized operator, enabling topological invariants for these complex systems.

Contribution

It introduces a new nondegeneracy condition for time-periodic solutions and proves the Fredholm property of the Floer operator in an infinite-dimensional setting.

Findings

Established compactness results for the PDE class.

Defined a new nondegeneracy notion for solutions.

Proved the Fredholm property of the linearized operator.

Abstract

By coupling a Hamiltonian mechanical system with a linear Hamiltonian field theory one obtains an infinite-dimensional Hamiltonian system with regularizing nonlinearity, where the underlying phase space is given by the product of a finite-dimensional symplectic manifold with an infinite-dimensional linear symplectic Hilbert space. After our compactness results we continue our program for defining a Floer homology theory for this class of infinite-dimensional Hamiltonian systems. Based on the presence of small divisors we introduce a new notion of nondegeneracy for time-periodic solutions which allows us to prove that the linearization of the nonlinear Floer operator is Fredholm when viewed as a map between suitable Sobolev space completions.