2013/07/31 by A. S. Nikolaev, Andrey Nikolaev · 4 citations
Mathematics · Physics and Astronomy · #Canonical coordinates #Formalism (music) #Holomorphic and Operator Theory #Laurent series #Moore–Penrose pseudoinverse #Operator (biology) #Perturbation (astronomy) #Quantum Mechanics and Non-Hermitian Physics #Resolvent #Resolvent formalism #Series (stratigraphy) #Spectral Theory in Mathematical Physics #math-ph #math.DS #math.MP #msc:37J40 #msc:70H09 #msc:70K45 #msc:70K65 #nlin.CD #nlin.SI
paper · pdf · doi:10.1007/s11232-015-0271-5
published in Theoretical and Mathematical Physics 182(3), 407-436 (Pleiades Publishing) · 33 pages. Version 3: Additional discussion of computational algorithm. The supplementary data files, containing demonstrations and large expressions are available from author
arxiv created 2014/08/26 · openalex publication_date 2015/03/01 · arxiv updated 2015/04/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
This work explores the structure of Poincare-Lindstedt perturbation series in Deprit operator formalism and establishes its connection to Kato resolvent expansion. A discussion of invariant definitions for averaging and integrating perturbation operators and their canonical identities reveals a regular pattern in a Deprit generator. The pattern was explained using Kato series and the relation of perturbation operators to Laurent coefficients for the resolvent of Liouville operator. This purely canonical approach systematizes the series and leads to the explicit expression for the Deprit generator in any perturbation order: G = - \mathsf SH Hi. Here, \mathsf SH is the partial pseudo-inverse of the perturbed Liouville operator. Corresponding Kato series provides a reasonably effective computational algorithm. The canonical connection of perturbed and unperturbed averaging operators allows for a description of ambiguities in the generator and transformed Hamiltonian, while Gustavson integrals turn out to be insensitive to normalization style. Non-perturbative examples are used for illustration.