Nonlinearized Perturbation Theories

DOI

10.2991/jnmp.1996.3.1-2.4How to use a DOI?

Abstract

A brief review is presented of the two recent perturbation algorithms. Their common idea lies in a not quite usual treatment of linear Schrödinger equations via nonlinear mathematical means. The first approach (let us call it a quasi-exact perturbation theory, QEPT) tries to get the very zero-order approximants already "almost exact", at a cost of leaving the higher-order computations more complicated. Technically, it constructs and employs solutions of certain auxiliary nonlinear systems of algebraic equations for the suitable zero-order couplings and energies. The second approach (a fixed-point perturbation theory, FPPT) pays more attention to the higher-order corrections. Its purpose lies in an improvement of construction of unperturbed propagators or, alternatively, of the closely related (so­called effective) finite-dimensional auxiliary Hamiltonians. On a technical level, it employs a factorization interpreted via certain nonlinear mappings and, finally, approximates some matrix elements by fixed points of these mappings. In a broad context of the "generalized Rayleigh-Schrödinger" perturbation strategy, both the prescriptions need just more summations over "intermediate states". QEPT defines its nondiagonal unperturbed propagators in terms of infinite continued fractions. FPPT introduces a further simplification via another finite system of nonlinear algebraic equations for fixed points. Thus, both the subsequent QE and FP steps of construction share the same mathematics.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Miloslav Znojil
PY  - 1996
DA  - 1996/05/01
TI  - Nonlinearized Perturbation Theories
JO  - Journal of Nonlinear Mathematical Physics
SP  - 51
EP  - 62
VL  - 3
IS  - 1-2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.1996.3.1-2.4
DO  - 10.2991/jnmp.1996.3.1-2.4
ID  - Znojil1996
ER  -