Temporal Dynamics in Perturbation Theory

TL;DR

This paper reformulates perturbation theory as a dynamical system, analyzing stability conditions for convergence of approximation sequences, and illustrates the approach with energy level calculations of an anharmonic oscillator.

Contribution

It introduces a dynamical systems perspective to perturbation theory, including stability criteria via multipliers and Lyapunov exponents, with practical application to quantum oscillators.

Findings

Multiple stability conditions for approximation convergence

Introduction of mapping multipliers and Lyapunov exponents

Successful calculation of anharmonic oscillator energy levels

Abstract

Perturbation theory can be reformulated as dynamical theory. Then a sequence of perturbative approximations is bijective to a trajectory of dynamical system with discrete time, called the approximation cascade. Here we concentrate our attention on the stability conditions permitting to control the convergence of approximation sequences. We show that several types of mapping multipliers and Lyapunov exponents can be introduced and, respectively, several types of conditions controlling local stability can be formulated. The ideas are illustrated by calculating the energy levels of an anharmonic oscillator.