Perturbations of integrable systems and Dyson-Mehta integrals

TL;DR

This paper develops an algebraic perturbation theory for quantum integrable systems, enabling linear algebra methods to compute corrections and classify perturbations, and provides algebraic calculations for generalized Dyson-Mehta integrals.

Contribution

It introduces an algebraic perturbation framework for integrable quantum systems and classifies perturbations via Lie algebra, facilitating explicit calculations of Dyson-Mehta integrals.

Findings

Algebraic perturbation theory for integrable systems developed.

Lie algebra classification of perturbations provided.

Explicit algebraic computation of Dyson-Mehta integral ratios achieved.

Abstract

We show that the existence of algebraic forms of quantum, exactly-solvable, completely-integrable $A-B-C-D$ and $G_2, F_4, E_{6,7,8}$ Olshanetsky-Perelomov Hamiltonians allow to develop the {\it algebraic} perturbation theory, where corrections are computed by pure linear algebra means. A Lie-algebraic classification of such perturbations is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. The approach also allows to calculate the ratios of a certain generalized Dyson-Mehta integrals algebraically, which are interested by themselves.