Symplectic non-squeezing in Hilbert space and discrete Schr\"odinger equations

TL;DR

This paper extends Gromov's symplectic non-squeezing theorem to infinite-dimensional Hilbert spaces and applies it to the flow of the discrete nonlinear Schr"odinger equation, advancing symplectic geometry in functional analysis.

Contribution

It generalizes symplectic non-squeezing to Hilbert spaces using complex discs and applies the result to discrete Schr"odinger equations.

Findings

Established symplectic non-squeezing in Hilbert spaces

Extended Gromov's results to infinite dimensions

Applied the theory to discrete nonlinear Schr"odinger flow

Abstract

We prove a generalization of Gromov's symplectic non-squeezing theorem for the case of Hilbert spaces. Our approach is based on filling almost complex Hilbert spaces by complex discs partially extending Gromov's results on existence of $J$-complex curves. We apply our result to the flow of the discrete nonlinear Schr\"odinger equation.