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

TL;DR

The paper addresses the challenge of high-order perturbation theory in systems with an infinite set of perturbations by introducing a partition-based framework that yields closed-form corrections at any order. It constructs a single matrix object $M^{(N)}$ from the hierarchical quantities $\Delta H^{(n)}$ and the resolvent $\Gamma$, with eigenvalue corrections given by $E^{(N)}=\langle0|M^{(N)}|0\rangle$ and eigenvector corrections by $|\psi^{(N)}\rangle=\Gamma M^{(N)}|0\rangle$, where $M^{(N)}$ sums over all ordered partitions of $N$. The main result expresses $M^{(N)}$ as an explicit sum over these partitions, enabling straightforward high-order calculations and recovering standard Rayleigh-Schrödinger theory in the single-perturbation limit. The approach situates itself among classical formalisms (Kato, Löwdin, Messiah, Soliverez, BP) while extending them to infinite perturbations and providing practical, implementable expressions (and accompanying code) for high-order perturbative analysis. This framework has potential to simplify complex perturbative computations and illuminate BCH-related expansions and phase-transition problems in statistical mechanics.

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.