Journal of Nonlinear Mathematical Physics

Volume 3, Issue 1-2, May 1996, Pages 51 - 62

Nonlinearized Perturbation Theories

Miloslav ZNOJIL
Corresponding Author
Miloslav ZNOJIL
Available Online 19 December 2006.
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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.
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Journal of Nonlinear Mathematical Physics
3 - 1
51 - 62
Publication Date
DOI to use a DOI?
Open Access
This is an open access article distributed under the CC BY-NC license.

Cite this article

AU  - Miloslav ZNOJIL
PY  - 2006
DA  - 2006/12
TI  - Nonlinearized Perturbation Theories
JO  - Journal of Nonlinear Mathematical Physics
SP  - 51
EP  - 62
VL  - 3
IS  - 1-2
SN  - 1402-9251
UR  -
DO  -
ID  - ZNOJIL2006
ER  -