Self-Similar Perturbation Theory

TL;DR

This paper introduces self-similar perturbation theory, a novel approach for analyzing the convergence and stability of approximation methods in complex physical problems, especially in quantum mechanics.

Contribution

It develops a stability analysis based on the method of multipliers and demonstrates its effectiveness through quantum-mechanical examples.

Findings

Provides criteria for convergence with limited approximation terms

Introduces a stability analysis method for perturbation algorithms

Shows improved decision-making in initial approximation selection

Abstract

A method is suggested for treating those complicated physical problems for which exact solutions are not known but a few approximation terms of a calculational algorithm can be derived. The method permits one to answer the following rather delicate questions: What can be said about the convergence of the calculational procedure when only a few its terms are available and how to decide which of the initial approximations of the perturbative algorithm is better, when several such initial approximations are possible? Definite answers to these important questions become possible by employing the self-similar perturbation theory. The novelty of this paper is in developing the stability analysis based on the method of multipliers and in illustrating the efficiency of this analysis by different quantum-mechanical problems.