Renormalization-group Resummation of Divergent Series of the Perturbative Wave Function of Quantum Anharmonic Oscillator

TL;DR

This paper applies renormalization group techniques to resum divergent perturbation series of the quantum anharmonic oscillator's wave function, revealing limitations in higher-order approximations compared to WKB results.

Contribution

It introduces a renormalization group approach to resum divergent series in quantum anharmonic oscillators and analyzes its effectiveness across different orders.

Findings

Resummed series is the cumulant of the naive perturbation series.

Agreement with WKB deteriorates beyond the fourth order.

Higher-order resummation does not improve accuracy in this context.

Abstract

The renormalization group method is applied for obtaining the asymptotic form of the wave function of the quantum anharmonic oscillator by resumming the perturbation series. It is shown that the resumed series is the cumulant of the naive perturbation series. Working out up to the sixth order and performing a further resummation proposed by Bender and Bettencourt, we find that the agreement with the WKB result becomes worse in the higher orders than the fourth at which the agreement is the best.