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 $n$ 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 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 $n$ 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.

Physical ReviewAug 15, 195728 citations

About Approximation of Convergence Superoperators in Quantum Perturbation Theory

Dorota Bielińska-Wąż, Ivana Paidarová, Ph. Durand

Perturbation methods are generally used for solving wave operator equations associated with the determination of effective Hamiltonians.In many cases the standard Rayleigh-Schrödinger and Brillouin-Wigner series either converge slowly or diverge.Therefore it is necessary to modify or to renormalize the standard wave equations.For that purpose derivative and convergence superoperators within the Ralyeigh-Schrödinger and Brillouin-Wigner formalisms were introduced.A new efficient approximation for convergence superoperators is investigated in this paper.Its application to a model system of N non-interacting molecules shows that this approximation can overcome convergence difficulties.Perturbation methods are generally used for solving wave operator equations associated with the determination of effective Hamiltonians.In many cases the standard Rayleigh-Schrödinger and Brillouin-Wigner series either converge slowly or diverge.Therefore it is necessary to modify or to renormalize the standard wave equations.For that purpose derivative and convergence superoperators within the Ralyeigh-Schrödinger and Brillouin-Wigner formalisms were introduced.A new efficient approximation for convergence superoperators is investigated in this paper.Its application to a model system of N non-interacting molecules shows that this approximation can overcome convergence difficulties.

Acta Physica Polonica AJun 1, 19970 citations

Dynamical Equations for the Wigner Functions

Jens Peder Dahl

Springer eBooksJan 1, 198320 citations

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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.