Generalisation of the explicit expression for the Deprit generator to Hamiltonians nonlinearly dependent on small parameter

TL;DR

This paper generalizes the explicit formula for the Deprit generator to Hamiltonians with nonlinear dependence on a small parameter, connecting perturbation series to Kato resolvent expansion and improving computational methods.

Contribution

It extends the Deprit perturbation formalism to nonlinear Hamiltonians and provides an explicit generator expression applicable at any order.

Findings

Derived explicit Deprit generator expression for nonlinear Hamiltonians

Identified regular patterns and non-uniqueness in the perturbation series

Demonstrated computational efficiency over classical methods

Abstract

This work explores a structure of the Deprit perturbation series and its connection to a Kato resolvent expansion. It extends the formalism previously developed for the Hamiltonians linearly dependent on perturbation parameter to a nonlinear case. We construct a canonical intertwining of perturbed and unperturbed averaging operators. This leads to an explicit expression for the generator of the Lie-Deprit transform in any perturbation order. Using this expression, we discuss a regular pattern in the series, non-uniqueness of the generator and normalised Hamiltonian, and the uniqueness of the Gustavson integrals. Comparison of the corresponding computational algorithm with classical perturbation methods demonstrates its competitiveness for Hamiltonians with a limited number of perturbation terms.