Algebraical Aspects of Perturbation Theories

Barry G. Adams

C. Vargas‐Hernández

Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm

This paper deals with perturbation theory for discrete spectra of linear operators. To simplify exposition we consider here self-adjoint operators. This theory is based on the Feshbach-Schur map and it has advantages with respect to the standard perturbation theory in three aspects: (a) it readily produces rigorous estimates on eigenvalues and eigenfunctions with explicit constants; (b) it is compact and elementary (it uses properties of norms and the fundamental theorem of algebra about solutions of polynomial equations); and (c) it is based on a self-contained formulation of a fixed point problem for the eigenvalues and eigenfunctions, allowing for easy iterations. We apply our abstract results to obtain rigorous bounds on the ground states of Helium-type ions.

A. V. Turbiner

We show that the existence of algebraic forms of exactly solvable A-B-C-D , G 2 , and F 4 Olshanetsky-Perelomov Hamiltonians allows one to develop algebraic perturbation theory, where corrections are computed by purely algebraic means. A classification of perturbations leading to such a perturbation theory based on the theory of representations of Lie algebras is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. Some examples are presented.

Konstantin V. Kazakov

A. M. Basharov

Abstract The algebraic resonance perturbation theory (ARPT) is formulated, general expressions of the theory are written down, by means of which effective Hamiltonians are concisely derived, interference interactions are described, and kinetic equations are obtained for various cases of interaction between electromagnetic radiation and quantum systems. The natural integration of the method of stochastic differential equations into ARPT and the physical justification of the principles for the selection of ARPT terms are indicated in the case of approximating the environment of an open system by delta-correlated noise. The inevitability of the emergence of interference interactions is emphasized due to the formulated ARPT requirements regarding the selection of only slowly time-varying terms in the Dirac picture into the effective Hamiltonian.