Optimized Perturbation Theory

Marcus Benghi Pinto

Marcus Benghi Pinto, Rudnei O. Ramos, Paulo José Sena dos Santos

Vyacheslav I. Yukalov

This is one of the two papers where the optimized perturbation theory was first formulated. The other paper is published in Theor. Math. Phys. 28, 652--660 (1976). The main idea of the theory is to reorganize the perturbative sequence by introducing control functions, defined by optimization conditions, so that the reorganized approximation sequence be convergent. In the present paper, the theory of perturbations is suggested for statistical systems in the absence of small interaction parameters. A new form is advanced for self-consistent conditions defining the optimal parameters for trial Green functions in iterating nonlinear propagator equations. Superharmonic, semiharmonic, and pseudoharmonic approximations for a molecular crystal are considered as examples.

Jean-Loı̈c Kneur, A. Neveu, Marcus Benghi Pinto

Improving perturbation theory via a variational optimization has generally produced in higher orders an embarrassingly large set of solutions, most of them unphysical (complex). We introduce an extension of the optimized perturbation method which leads to a drastic reduction of the number of acceptable solutions. The properties of this method are studied and it is then applied to the calculation of relevant quantities in different ${\ensuremath{\phi}}^{4}$ models, such as the anharmonic oscillator energy levels and the critical Bose-Einstein condensation temperature shift $\ensuremath{\Delta}{T}_{c}$ recently investigated by various authors. Our present estimates of $\ensuremath{\Delta}{T}_{c}$, incorporating the most recently available six and seven loop perturbative information, are in excellent agreement with all the available lattice numerical simulations. This represents a very substantial improvement over previous treatments.

Walter Greiner

Lorenzo Chua Chan, A. Villanueva

The paper discusses how the variationally inspired perturbation theory (VIPT) scheme of approximation in quantum mechanics can be improved convergencewise if one uses for variational trial functions the perturbative series with variational parameter, so that one is effectively doing variational calculations directly on the high-order perturbative series for energy. The result optimizes the high-order energy directly and thus represents a significant improvement over the VIPT procedure. When applied to a double potential in which even the VIPT is badly divergent, we saw that the result is still very much convergent.