# Hamiltonian Floer theory for nonlinear Schr\"odinger equations and the small divisor problem

## TL;DR

This paper establishes the existence of infinitely many time-periodic solutions for nonlinear Schrödinger equations by applying Hamiltonian Floer theory and addressing the small divisor problem through elliptic and Diophantine approximation methods.

## Contribution

It extends Hamiltonian Floer theory to infinite-dimensional PDEs and provides a novel approach to the small divisor problem in this context.

## Key findings

- Proves existence of infinitely many solutions
- Develops a method to handle small divisor issues
- Bridges Floer theory with nonlinear PDE analysis

## Abstract

We prove the existence of infinitely many time-periodic solutions of nonlinear Schr\"odinger 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.