Perturbations of integrable systems

Alexander Turbiner (turbiner@nucleu.unam.mx)
Instituto de Ciencias Nucleares
UNAM
Apartado Postal 70-543 04510
Mexico D.F. MEXICO

Abstract
Olshanetsky-Perelomov quantum Hamiltonians are integrable and also exactly-solvable. They admit algebraic forms being represented as linear differential operators with polynomial coeffs. The algebraic form allows to find a quite general class of non-trivial perturbations for which one can develop a constructive, "algebraic perturbation theory", where all corrections are found by algebraic means. These perturbations are classified and some many-body anharmonic oscillators are among the perturbed problems. Fock space formalism is presented, which gives rise an isospectral, polynomiality-of-eigenfunctions-preserving correspondence between integrable continuous systems and a certain finite-difference eqs on uniform and exponential lattices. Algebraic perturbation theory allows to study perturbations of continuous and discrete systems in the same time.