Perturbation Theory and Control in Classical or Quantum Mechanics by an Inversion Formula

TL;DR

This paper develops an inversion formula for perturbations in classical and quantum Hamiltonian systems, enabling simplification of perturbed systems and applications to control theory and quantum adiabatic transformations.

Contribution

It introduces a new inversion formula for perturbations in Hamiltonian systems, applicable to both classical and quantum mechanics, with bounds and practical examples.

Findings

Derived an explicit expression for the transformation simplifying perturbed systems.

Provided bounds for perturbation size to ensure invertibility.

Applied the formula to control theory and quantum adiabatic processes.

Abstract

We consider a perturbation of an ``integrable'' Hamiltonian and give an expression for the canonical or unitary transformation which ``simplifies'' this perturbed system. The problem is to invert a functional defined on the Lie- algebra of observables. We give a bound for the perturbation in order to solve this inversion. And apply this result to a particular case of the control theory, as a first example, and to the ``quantum adiabatic transformation'', as another example.