KAM for KG on $\mathbb S^2$ and for the quantum harmonic oscillator on $\mathbb R^2$

TL;DR

This paper develops a KAM theorem applicable to the Klein-Gordon equation on the sphere and the quantum harmonic oscillator on the plane, demonstrating persistence of quasi-periodic solutions under perturbations.

Contribution

It introduces an abstract KAM theorem tailored for these specific PDEs with regularizing nonlinearities, advancing the understanding of their stability.

Findings

Proves the existence of quasi-periodic solutions for the Klein-Gordon equation on 2^2.

Establishes stability results for the quantum harmonic oscillator on 2^2.

Provides a framework for applying KAM theory to PDEs on curved and flat spaces.

Abstract

In this paper we prove an abstract KAM theorem adapted to the Klein Gordon equation on the sphere $\mathbb S^2$ and for the quantum harmonic oscillator on $\mathbb R^2$ with regularizing nonlinearity.