Renormalization-group Resummation of a Divergent Series of the Perturbative Wave Functions of Quantum Systems

TL;DR

This paper applies renormalization-group techniques to resum divergent perturbative series of quantum wave functions, improving their asymptotic accuracy and connecting to nonperturbative methods.

Contribution

It introduces a RG-based resummation method for divergent series of quantum wave functions and relates it to the delta-expansion approach.

Findings

Resummed series yields the cumulant of the naive perturbation series.

Reorganization reproduces the correct asymptotic form at infinity.

Method connects perturbative and nonperturbative RG approaches.

Abstract

The perturbative renormalization group(RG) equation is applied to resum divergent series of perturbative wave functions of quantum anharmonic oscillator. It is found that the resummed series gives the cumulant of the naive perturbation series. It is shown that a reorganization of the resummed series reproduce the correct asymptotic form of the wave function at $x\to \infty$ when the perturbation expansion is stopped at the fourth order. A brief comment is given on the relation between the present method and the delta-expansion method, which is based on a kind of a nonperturbative RG equation.