An explicit formula for perturbation theory at any order with infinitely many perturbations

TL;DR

This paper introduces a systematic, explicit formula for high-order perturbation theory involving infinitely many perturbations, simplifying calculations and unifying eigenvalue and eigenvector corrections in a single matrix framework.

Contribution

It presents a novel, explicit formula based on integer partitions that handles arbitrary order and infinite perturbations in a unified manner.

Findings

Provides a closed-form formula for perturbation series at any order.

Includes an infinite number of perturbations seamlessly.

Reduces to standard perturbation theory in the single perturbation limit.

Abstract

We provide a systematic formula, in terms of integer partitions, that generates perturbation theory explicitly at an arbitrary order. Our approach naturally includes an infinite number of perturbations and uses a single matrix equation that contains the information for both the eigenvalue and eigenvector corrections. The formula reduces to the standard case of one perturbation in the appropriate limit. This formulation streamlines the derivations that are traditionally tedious in perturbation theory, facilitating high-order calculations.