Rescaled Perturbation Theory

TL;DR

This paper introduces a non-perturbative rescaled perturbation theory that formulates exact differential equations in coupling strength, enabling accurate calculations across weak to strong coupling regimes in quantum systems.

Contribution

The paper presents a novel rescaled perturbation method that surpasses traditional weak coupling expansions by using differential equations, applicable to non-Borel summable series and time-dependent systems.

Findings

Accurately computes energy eigenvalues and wave functions from weak to strong coupling.

Successfully applies to quantum anharmonic oscillator and double well potential.

Works well for systems with divergent and non-Borel summable series.

Abstract

A non-perturbative method which can go beyond the weak coupling perturbation theory is introduced. Essential idea is to formulate a set of exact differential equations as a function of the coupling strength $g$. Unlike other resummation in which information of the higher order terms is necessary, we only need a leading order perturbative formula in every step to reach the large value of $g$. The method is applied to the quantum anharmonic oscillator and quantum double well potential in one dimension. Both are known to have divergent series in the weak coupling perturbation and the latter is not Borel summable. Our method is shown to work well from the weak coupling to the strong coupling for the energy eigenvalues and the wave functions. The method is also applied successfully to the system with time-dependent external field.