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arXiv:2511.05153 · Physical review. B./Physical review. B

Explicit formula for perturbation theory at any order with infinitely many perturbations

J. M. Jones, M. W. Long

Published Feb 24, 2026• 0 citations
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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.

DOI
10.1103/2m8h-hvkj

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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.
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.
The standard textbook formulas for the perturbation corrections are somewhat cumbersome for a systematic calculation of sufficiently high order. In this book we show several alternative strategies that are easily programmable for numerical or algebraic calculation. We are mainly concerned with the derivation of exact perturbation corrections, and therefore concentrate on sufficiently simple nontrivial models having physical application. However, some of the algorithms discussed in this book are also suitable for numerical calculation.
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 nondegenerate perturbation theory is studied using the adiabatic-theorem formalism. A new factorization of the adiabatic time-evolution operator analogous to that obtained by Morita in his derivation of the linked-cluster theorem is given by carrying out a series of intermediate representations. This leads to a new expansion for the perturbed eigenvectors \ensuremath{\Vert}\ensuremath{\alpha}〉 and energies ${E}_{\ensuremath{\alpha}}$ which are analytically expressed with respect to two matrix series scrV(n){\mathrm{scrV}}^{(n)}scrV(n) and E(n){E}^{(n)}E(n). The nth term of each series (scrV(n){\mathrm{scrV}}^{(n)}scrV(n) and E(n){E}^{(n)}E(n)) includes perturbational contributions of power p, p\ensuremath{\ge}n. An explicit recurrence, scrV(n){\mathrm{scrV}}^{(n)}scrV(n)=V(${\mathrm{scrV}}^{n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$), E(n){E}^{(n)}E(n)=E(${\mathrm{scrV}}^{(n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$) is derived and allows us to calculate these matrices to an arbitrary iteration order (n). This is a central feature of our approach since the eigenvectors and eigenvalues can be generated, to this arbitrary order, by a simple numerical program.
Exactly solvable models in quantum mechanics are very rare.However, perturbation theory is a powerful tool to research the complex quantum systems and is widely used in theoretical physics and chemistry.For the particle weakly perturbed by the surrounding environment, its energy eigenstate and eigenvalue are separately calculated up to the first-order and second-order perturbation approximations in the undergraduate textbook of quantum mechanics.Whereas in reality, we have to calculate the high-order perturbation correction terms occasionally.Along the same lines of the low-order perturbation correction terms of energy eigenvalues and eigenstate for stationary state perturbation theory in the non-degenerate case, the third-order and fourth-order perturbation correction terms are strictly attained in this paper.Moreover, it points out the essence of perturbation theory and provides its derivation ideas.Students are no longer afraid of tedious derivation and can derive any-order perturbation correction.These results can serve as an extension and supplement for the quantum mechanics textbooks.Furthermore, it is very helpful for students to deeply understand quantum perturbation theory.
The solution of the quantum-mechanical eigenvalue problem is discussed for cases when a series of approximate eigenfunctions is known. If these ``unperturbed'' states are divided into two classes, a perturbation formula is derived giving the influence of one class of states on the other in the final solution. The formula contains as special cases: (i) the Schrödinger-Brillouin formula for the eigenvalue of a nondegenerate state, (ii) a new simple formula for treating a class of degenerate states, and (iii) the splitting of the secular equation in cases where the system naturally consists of two independent parts in mutual interaction.