KAM for the quantum harmonic oscillator

TL;DR

This paper proves an abstract KAM theorem for infinite-dimensional Hamiltonian systems and applies it to demonstrate the existence of quasi-periodic solutions and reducibility in 1D nonlinear Schrödinger equations with harmonic potentials.

Contribution

It extends existing KAM results to broader classes of infinite-dimensional systems and provides new applications to Schrödinger equations with harmonic potentials.

Findings

Existence of many quasi-periodic solutions for 1D nonlinear Schrödinger equations.

Proven reducibility of Schrödinger equations with harmonic potential and time-quasi-periodic perturbations.

Extended KAM theory applicable to infinite-dimensional Hamiltonian systems.

Abstract

In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. P\"oschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schr\"odinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schr\"odinger equations with the harmonic potential and a quasi periodic in time potential.