Transformation of perturbative series into complex phases and elimination of secular divergences in time-dependent perturbation theory in quantum mechanics

TL;DR

This paper introduces a method to transform perturbative series into exponential functions to eliminate secular divergences in time-dependent quantum perturbation theory, validated by exact solutions of three models.

Contribution

The paper presents a novel transformation technique that re-sums perturbative series into exponential forms, effectively removing secular divergences in quantum dynamics.

Findings

Excellent agreement with exact solutions in three models

Secular divergences are effectively eliminated

Method improves long-time accuracy of perturbative calculations

Abstract

The difficulty that the probabilities infinitely increase with time as time is long enough in time-dependent perturbation theory for some quantum systems is resolved by means of simply transforming the perturbative series into natural exponential functions of the re-summed perturbative series. Three exactly solvable models are taken to check our new formulation, and excellent agreements with the exact solution are achieved.