Perturbation series in operator and superoperator formalism. I

K. Lendi

Jean‐Paul Montagner, David Mainprice

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.

Bryce Seligman DeWitt

Roberto C. Bochicchio, Horacio Grinberg

A direct approach for the generalized reduced Liouville equation of motion decoupling problem associated with open quantum systems within the superoperator formalism is presented. The procedure is based on inversion of the perturbation series for the energy representation of the density operator so as to obtain one for the proper self-energy fields that emerge as a consequence of the analytic character of the associated spectral representation. Thus, the perturbation series that arises from the iteration of the energy-dependent matrix elements hierarchy involved in the statistical operator allows, upon further expansion of the inverse of such series, to get formally exact expressions for the corrections to all orders of the self-energy fields. The lower order corrections of these fields are discussed in terms of resonant and nonresonant contributions. The present approach provides matrix equations that show the close relation between the environment effects represented by the self-energy fields and the relaxation kernel that drives the system–reservoir interaction. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 280–290, 2000

Jeremy M. Moix, Eli Pollak, Jiushu Shao

A generalized time-dependent perturbation theory is derived for superoperators. Instead of using the ``standard'' breakup of the Hamiltonian into a known zeroth order term and a correction, we use the approximate superpropagator to define the correction superoperator which is then used to obtain a series representation of the exact Liouville operator. The theory reduces to known limits and may be used for a perturbation expansion of classical Wigner and Husimi dynamics as well as for recent phase-space-based semiclassical approximations. The theory is demonstrated for a model quartic potential.