Floer homology for Hamiltonian PDEs: Fredholm theory
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.
Keywords
Cite
@article{arxiv.2107.14074,
title = {Floer homology for Hamiltonian PDEs: Fredholm theory},
author = {Oliver Fabert and Niek Lamoree},
journal= {arXiv preprint arXiv:2107.14074},
year = {2021}
}
Comments
23 pages; added Hamiltonian particle-field systems such as coupled Maxwell-Lorentz equations and other examples as motivation, generalized set-up, more details, small corrections, removed a redundancy