Quantum integrable systems and representations of Lie algebras

TL;DR

This paper constructs quantum integrals for specific many-body Hamiltonians using Lie algebra representations, providing new proofs and expressions for eigenfunctions in integrable quantum systems.

Contribution

It introduces a novel approach linking Lie algebra Casimirs to quantum integrals and offers new proofs and formulas for eigenfunctions in integrable models.

Findings

Constructed quantum integrals from Lie algebra Casimirs.

Provided a new proof of the Olshanetsky-Perelomov theorem.

Derived new expressions for eigenfunctions as traces of intertwining operators.

Abstract

In this paper the quantum integrals of the Hamiltonian of the quantum many-body problem with the interaction potential K/sinh^2(x) (Sutherland operator) are constructed as images of higher Casimirs of the Lie algebra gl(N) under a certain homomorphism from the center of U(gl(N)) to the algebra of differential operators in N variables. A similar construction applied to the affine gl(N) at the critical level k=-N defines a correspondence between higher Sugawara operators and quantum integrals of the Hamiltonian of the quantum many-body problem with the potential equal to constant times the Weierstrass function. This allows one to give a new proof of the Olshanetsky-Perelomov theorem stating that this Hamiltonian defines a completely integrable quantum system. We also give a new expression for eigenfunctions of the quantum integrals of the Sutherland operator as traces of intertwining operators between certain representations of gl(N).