On the convergence of the usual perturbative expansions

TL;DR

This paper investigates the convergence properties of power series expansions for energy eigenvalues in quantum anharmonic oscillators, especially focusing on quasi-exactly solvable potentials and their unique convergence behaviors.

Contribution

It offers new insights into the convergence of perturbative expansions for specific quantum potentials and introduces techniques for analyzing more general cases.

Findings

Identifies finite radius of convergence for certain potentials

Provides methods to analyze convergence in quasi-exactly solvable cases

Highlights differences from general quantum mechanical series expansions

Abstract

The study of the convergence of power series expansions of energy eigenvalues for anharmonic oscillators in quantum mechanics differs from general understanding, in the case of quasi-exactly solvable potentials. They provide examples of expansions with finite radius and suggest techniques useful to analyze more generic potentials.