A semi-classical K.A.M. theorem

TL;DR

This paper proves a semi-classical K.A.M. theorem for integrable systems on the torus, establishing conjugacy to normal form and constructing quasimodes for perturbed operators.

Contribution

It introduces a method to conjugate perturbed semi-classical integrable systems to a normal form, enabling the construction of quasimodes.

Findings

Successful conjugation to normal form for perturbed systems

Construction of a large number of quasimodes

Extension of K.A.M. theory to semi-classical pseudodifferential operators

Abstract

We consider a semi-classical completely integrable system defined by a $\hbar$-pseudodifferential operator $\hat{H}$ on the torus $\mathbb{T}^{d}$. In order to study perturbed operators of the form $\hat{H}+\hbar^{\kappa}\hat{K}$, where $\hat{K}$ is an arbitrary pseudodifferential operator and $\kappa>0$, we prove the conjugacy to a suitable normal form. This is then used to construct a large number of quasimodes.