Volume 3, Issue 1-2, May 1996, Pages 51 - 62
Nonlinearized Perturbation Theories
- Miloslav ZNOJIL
- Corresponding Author
- Miloslav ZNOJIL
Available Online 19 December 2006.
- https://doi.org/10.2991/jnmp.1996.3.1-2.4How to use a DOI?
- 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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TY - JOUR 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 - 1776-0852 UR - https://doi.org/10.2991/jnmp.1996.3.1-2.4 DO - https://doi.org/10.2991/jnmp.1996.3.1-2.4 ID - ZNOJIL2006 ER -