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

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.