Convergence of Quantum Lindstedt Series and Semiclassical Renormalization

Abstract

In this work we consider the KAM renormalizability problem for small pseudodifferential perturbations of the semiclassical isochronous transport operator with Diophantine frequencies on the torus. Assuming that the symbol of the perturbation is real analytic and globally bounded, we prove convergence of the quantum Lindstedt series and describe completely the set of semiclassical measures and quantum limits of the renormalized system. Each of these measures is given by symplectic deformation of the Haar measure on an invariant torus for the unperturbed classical system.