Convergence of Quantum Lindstedt Series and Semiclassical Renormalization

TL;DR

This paper proves the convergence of the quantum Lindstedt series for certain semiclassical operators and characterizes the resulting quantum measures, advancing understanding of quantum-classical correspondence in perturbed integrable systems.

Contribution

It establishes the convergence of the quantum Lindstedt series for semiclassical isochronous systems with Diophantine frequencies and describes the quantum limits explicitly.

Findings

Convergence of the quantum Lindstedt series is proven.

Quantum limits are characterized as symplectic deformations of Haar measures.

The set of semiclassical measures is fully described for the renormalized system.

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.