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.
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