Hamiltonian Floer Theory for Nonlinear Schrödinger Equations and the Small Divisor Problem

Abstract

We prove the existence of infinitely many time-periodic solutions of nonlinear Schrödinger equations using pseudo-holomorphic curve methods from Hamiltonian Floer theory. For the generalization of the Gromov–Floer compactness theorem to infinite dimensions, we show how to solve the arising small divisor problem by combining elliptic methods with results from the theory of diophantine approximations.

Cuplength estimates for periodic solutions of Hamiltonian particle-field systems

“…Note that this is an infinite-dimensional analogue of the perturbed Cauchy-Riemann equation used to define Floer homology for general symplectomorphisms in [6]. The following main theorem of this paper is an analogue of [8,Theorem 10.4], see also [7,Theorem 3.4].…”

“…In analogy with our work in [7] and [8], it is the goal of this paper to prove the existence of T -periodic solutions of…”

“…In order to prove Theorem 2.1, we essentially combine our work in [7] and [8] on pseudoholomorphic curves for infinite-dimensional Hamiltonian systems with the celebrated paper [5] on pseudoholomorphic curves in cotangent bundles in order to prove an existence result for Floer curves in T * Q × H.…”

“…Apart from the fact that the underlying phase space M = T * Q × H is infinitedimensional, as in [7], [8] we first need to deal with the fact that the field Hamiltonian…”

“…As in [2], [5], [14] and [7], [8] we follow the idea of A. Floer in [9] to study flow lines of the gradient ∇ A Ḡ φ of the action functional with respect to the L 2 -metric on H 1 φ (R, T * Q × H) given by the canonical Riemannian metric •, • on M = T * Q × H. The reason why Floer preferred to choose the L 2 -gradient over the more natural H 1 -gradient follows from the observation that the gradient flow equation…”