A semi-classical K.A.M. theorem

Abstract

We consider a semi-classical completely integrable system defined by a $\hslash$-pseudodifferential operator $\hat{H}$ on the torus ${\mathbb{T}}^d$. In order to study perturbed operators of the form $\hat{H}+{\hslash }^{\kappa }\hat{K}$, 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.

One mistake in Proposition 15 corrected, introduction shortened, typographics errors fixed, bibliography updated