Canonical forms for perturbations of the harmonic oscillator

TL;DR

This paper studies perturbations of the 2D harmonic oscillator and similar systems, demonstrating they can be transformed into a canonical form and establishing a quantum normal form linked to spectral data.

Contribution

It introduces a method to convert certain perturbed harmonic oscillators into a canonical form and constructs a corresponding quantum normal form based on spectral information.

Findings

Perturbed systems are isomorphic to a toric system (Birkhoff canonical form).

Existence of a quantum normal form determined by spectral data.

Applicable to a class of dynamical systems similar to the harmonic oscillator.

Abstract

We consider a class of perturbations of the 2D harmonic oscillator, and of some other dynamical systems, which we show are isomorphic to a function of a toric system (a Birkhoff canonical form). We show that for such systems there exists a quantum normal form as well, which is determined by spectral data.