A semi-classical K.A.M. theorem

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.