Quantum Many--Body Problems and Perturbation Theory

TL;DR

This paper develops an algebraic perturbation theory for certain exactly-solvable quantum many-body Hamiltonians, enabling algebraic computation of corrections and classification of perturbations via Lie algebra representation theory.

Contribution

It introduces an algebraic perturbation framework for solvable many-body Hamiltonians based on Lie algebra representations, allowing explicit anharmonic problem analysis.

Findings

Established algebraic perturbation methods for specific Hamiltonians.

Classified perturbations compatible with algebraic perturbation theory.

Provided examples demonstrating the method's application.

Abstract

We show that the existence of algebraic forms of exactly-solvable $A-B-C-D$ and $G_2, F_4$ Olshanetsky-Perelomov Hamiltonians allow to develop the {\it algebraic} perturbation theory, where corrections are computed by pure algebraic means. A classification of perturbations leading to such a perturbation theory based on representation theory of Lie algebras is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. Some examples are presented.