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arXiv:2412.05166 · Physical Review A

Non-Hermitian generalization of Rayleigh-Schrödinger perturbation theory

Wei-Ming Chen, Yen-Ting Lin, Chia-Yi Ju

Published Feb 7, 2025• 6 citations
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Abstract

The authors develop a perturbation theory for non-Hermitian systems based on a geometric formalism that yields relatively simple recursive formulas for the perturbed eigenvectors and eigenvalues. The authors provide an example and show that the method proposed reduces to the standard Rayleigh-Schr\"odinger perturbation theory in the Hermitian regime.

DOI
10.1103/physreva.111.022211

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The authors develop a perturbation theory for non-Hermitian systems based on a geometric formalism that yields relatively simple recursive formulas for the perturbed eigenvectors and eigenvalues. The authors provide an example and show that the method proposed reduces to the standard Rayleigh-Schr"odinger perturbation theory in the Hermitian regime.
We develop a Rayleigh--Schrödinger-like perturbation theory for non-Hermitian quantum systems at an exceptional point of order NNN. Working in the Jordan basis of the unperturbed Hamiltonian and employing a Puiseux expansion of the perturbed eigenvalues and eigenstates, we derive explicit recursion relations for the expansion coefficients. The corrections to the unperturbed eigenvalue in the Puiseux expansion govern the splitting near the exceptional point; the first two are obtained iteratively in two equivalent forms. One is given in terms of the perturbation Hamiltonian in the Jordan basis, and the other in terms of the generator that drives the eigenvalue evolution with respect to the perturbation. The latter constitutes the exceptional-point counterpart of a geometric perturbation method recently developed for the non-exceptional-point regime. Both representations are verified explicitly for the N=2N = 2N=2 and N=3N = 3N=3 cases.
The authors develop a simple alternative approach to perturbation theory in one-dimensional nonrelativistic quantum mechanics. The formulae for the energy shifts and wavefunctions do not involve cumbersome sums over intermediate states which appear in the usual Rayleigh-Schrodinger expansion. Unlike standard logarithmic perturbation theory, the approach does not utilize taking the logarithm of the wavefunction, and is therefore applicable in the same form to both the ground state and excited bound states.
Non-Hermitian quantum-Hamiltonian-candidate combination H λ of a non-Hermitian unperturbed operator H = H 0 with an arbitrary "small" non-Hermitian perturbation λ W is given a mathematically consistent unitary-evolution interpretation. The formalism generalizes the conventional constructive Rayleigh-Schrödinger perturbation expansion technique. It is sufficiently general to take into account the well known formal ambiguity of reconstruction of the correct physical Hilbert space of states. The possibility of removal of the ambiguity via a complete, irreducible set of observables is also discussed.
A perturbation theory is described which leads to two alternative methods for calculating the eigenvalues and normalized eigenstates of a perturbed Hamiltonian, the results being given to all orders in the perturbation. The expansions obtained are formulated in terms of two sets of partial sums which have simple, easily characterized forms, and they combine some of the advantages of both the Rayleigh-Schr\"odinger and Brillouin-Wigner perturbation methods while avoiding some of the difficulties of each. In particular, one form of the perturbation expansions given here will in most cases of interest have a larger radius of convergence than the Rayleigh-Schr\"odinger series. As is usual for infinite-order treatments, the explicit form of the final result applies only to nondegenerate levels. However, some of the intermediate relations obtained in the derivation are of general validity. Since certain classes of unlinked diagrams appear in the expansions, the method described here is not suitable for application to systems containing a large number of particles.
A formulation of Schr\"odinger perturbation theory is developed that gives a unified treatment of non-degenerate and degenerate cases, is unique, and has a nonzero radius of convergence under very general conditions. Two alternative procedures are given for finding perturbed eigenvectors, one of which is simpler for the nondegenerate case or for small finite degeneracy, the other simpler for infinite or large finite degeneracy. The low-order terms in the perturbation expansions of quantities used in applications are given. The perturbation theory formulated in this paper has the following advantages over the conventional Schr\"odinger and Brillouin-Wigner perturbation theories: (i) When the convergence criterion is satisfied, bounds on the error made in replacing an appropriate infinite perturbation series by its first nnn terms can be obtained. (ii) For the case of degeneracy, the conventional Schr\"odinger perturbation theory can break down under conditions to which the convergence of the perturbation theory developed in this paper are insensitive. (iii) There is no implicit dependence on the eigenvalue, such as appears in the Brillouin-Wigner perturbation theory. (iv) For the case of degeneracy, statistical information about the distribution of certain eigenvalues can be obtained without finding the individual eigenvalues. (v) The theory is applicable to a wider class of problems than the conventional Schr\"odinger and Brillouin-Wigner perturbation theories.
A special nondegenerate perturbation theory is developed for solving the "nonlinear Schr\"odinger equation" of the type $[\stackrel{^}{H}+\stackrel{^}{V}(\ensuremath{\Psi})]\ensuremath{\Psi}=E\ensuremath{\Psi}$, which arises, e.g., from the study of the environmental effects on molecules. The interaction operator is taken as $\stackrel{^}{V}(\ensuremath{\Psi})=\stackrel{^}{A}〈\ensuremath{\Psi}|\stackrel{^}{B}|\ensuremath{\Psi}〉$. The in vacuo problem $\stackrel{^}{H}\ensuremath{\Phi}=\ensuremath{\epsilon}\ensuremath{\Phi}$ is considered the zeroth-order description. For the perturbed energies and wave functions recursive formulas are derived at any order of the perturbation. The formulas obtained are compared to the usual Rayleigh-Schr\"odinger perturbation theory where $\stackrel{^}{V}$ is supposed to be independent of $\ensuremath{\Psi}$. The formal results are applied to the case of the Onsager cavity model.