Secular Terms in Dyson Series to All-Orders of Perturbation

TL;DR

This paper demonstrates a systematic method to eliminate secular terms in the Dyson series for quantum systems, enabling all-order perturbative calculations of energy levels and decay rates.

Contribution

It introduces an improved perturbation method that removes secular divergences to all orders in Dyson series for quantum systems.

Findings

Secular terms can be systematically removed to all orders.

Explicit recurrence relation for secular term removal.

Method to compute energy eigenvalues and decay rates to all orders.

Abstract

In classical and quantum systems, perturbation of an evolution equation is often invalidated by secular terms which diverge at late times. The diverging behavior of evolution can be remedied by various techniques of resumma- tion such as renormalization group or multi-scale analysis. In this paper, we prove that, in a generic quantum mechanical system, secular terms can be systematically removed to all orders in the Dyson series by the method of improved (renormalized) perturbation. A recurrence relation to provide an explicit method to remove the secular terms is given. As a byproduct, we give a simple method to obtain energy eigenvalues and decay rates to all orders of perturbation.