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Annalen der Physik

Superharmonic Approximation for Crystal

Vyacheslav I. Yukalov

Published Jan 1, 1980• 18 citations
Publisher

Abstract

Abstract The localized approximation for a quantum crystal considered earlier by the author is taken here as the initial step of an iteration, and the higher approximations for the propagators and thermodynamic functions are calculated. This generalization is based on a new type of a double iterative procedure for the equation of motion. The method is illustrated with a three‐dimensional crystal.

DOI
10.1002/andp.19804920303

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Journal of Mathematical Physics•Feb 1, 1973•34 citations
Abstract The localized approximation for a quantum crystal considered earlier by the author is taken here as the initial step of an iteration, and the higher approximations for the propagators and thermodynamic functions are calculated. This generalization is based on a new type of a double iterative procedure for the equation of motion. The method is illustrated with a three‐dimensional crystal.
Abstract We present here a mathematical basis of the iterative procedure developed earlier by the author for strongly anharmonic crystals. We generalize the approach to the case of any strongly nonideal nonuniform system. The convergence of this procedure is defined as a convergence in the mean with respect to averages of observable‐quantity operators. The definitions yield self‐consistent conditions for a zero‐order iteration in a natural way. Divergences in high‐order iterations are considered and two ways to eliminate them are proposed. The method is applied to a strongly‐anisotropic strongly‐anharmonic quantum crystal.
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.