Non-Hermitian Generalization of Rayleigh-Schr\"odinger Perturbation Theory

TL;DR

This paper extends Rayleigh-Schr"odinger perturbation theory to non-Hermitian quantum systems using a geometric formalism, enabling iterative calculations of eigenstates and eigenvalues beyond the Hermitian case.

Contribution

It introduces a geometric framework for non-Hermitian perturbation theory, generalizing the classical approach and revealing connections to Girard-Newton formulas.

Findings

Framework computes perturbative corrections to any order.

Recursion equations resemble Girard-Newton formulas.

Reduces to standard theory in Hermitian limit.

Abstract

While perturbation theories constitute a significant foundation of modern quantum system analysis, extending them from the Hermitian to the non-Hermitian regime remains a non-trivial task. In this work, we generalize the Rayleigh-Schr\"odinger perturbation theory to the non-Hermitian regime by employing a geometric formalism. This framework allows us to compute perturbative corrections to eigenstates and eigenvalues of Hamiltonians iteratively to any order. Furthermore, we observe that the recursion equation for the eigenstates resembles the form of the Girard-Newton formulas, which helps us uncover the general solution to the recursion equation. Moreover, we demonstrate that the perturbation method proposed in this paper reduces to the standard Rayleigh-Schr\"odinger perturbation theory in the Hermitian regime.