KAM for high-dimensional nonlinear quantum harmonic oscillator

TL;DR

This paper applies a KAM theorem to demonstrate the existence of many quasi-periodic solutions in high-dimensional nonlinear quantum harmonic oscillators, using a novel approach based on the decay of Hessian matrices.

Contribution

It extends the classical KAM theory to high-dimensional nonlinear quantum oscillators with multiple normal frequencies, introducing a new decay structure approach.

Findings

Existence of many quasi-periodic solutions.

Development of an abstract infinite-dimensional KAM theorem.

Application of decay structure of Hessian matrices.

Abstract

In this paper, we study high-dimensional nonlinear quantum harmonic oscillator equation. We show the equation admits many time quasi-periodic solutions by establishing an abstract infinite dimensional KAM theorem with multiple normal frequencies. The proof is based on the classical KAM scheme, and the key is a decaying structure of Hessian matrices of Hamiltonian functions.