Non-Hermitian Generalization of Rayleigh-Schrödinger Perturbation Theory

TL;DR

This work addresses the challenge of extending Rayleigh–Schrödinger perturbation theory to non-Hermitian Hamiltonians by formulating a geometric perturbation framework with an emergent dimension in parameter space. It introduces a $q$-generator $K(q,t)$ constrained by $i \partial_t K - i \partial_q H + [K,H] = 0$, decomposed as $K_1 t + K_0$, and derives recursion relations for eigenstate corrections $|n^{(k)}\rangle$ expressed via dual non-commutative Bell polynomials, alongside a gauge-invariant recursion for eigenvalues $h_n^{(k)}$. In the Hermitian limit, the method reduces to standard RS perturbation theory, while remaining applicable to non-Hermitian, $\mathcal{PT}$-symmetric, and pseudo-Hermitian systems; an explicit toy example demonstrates second-order non-Hermitian corrections and the insensitivity of eigenvalues to gauge parameters. The approach enriches perturbation theory with a structured, algebraic backbone and points toward connections with the quantum bootstrap and broader non-Hermitian physics.

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-Schrödinger 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-Schrödinger perturbation theory in the Hermitian regime.