Algebraic integrability of Schrödinger operators and representations of Lie algebras

Details Algebraic integrability of Schrödinger operators and representations of Lie algebras Algebraic integrability of Schrödinger operators and representations of Lie algebras

1994-03-22

arxiv.org/abs/hep-th/9403135v1

Physics

High Energy Physics

High Energy Physics - Theory

19 pages

Scientific paper

In this paper we study integrability and algebraic integrability properties of certain matrix Schr\"odinger operators. More specifically, we associate such an operator (with rational, trigonometric, or elliptic coefficients) to every simple Lie algebra g and every representation U of this algebra with a nonzero but finite dimensional zero weight subspace. The Calogero-Sutherland operator is a special case of this construction. Such an operator is always integrable. Our main result is that it is also algebraically integrable in the rational and trigonometric case if the representation U is highest weight. This generalizes the corresponding result for Calogero-Sutherland operators proved by Chalyh and Vaselov. We also conjecture that this is true for the elliptic case as well, which is a generalization of the corresponding conjecture by Chalyh and Vaselov for Calogero-Sutherland operators.

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