Algebraic integrability of Schrödinger operators and representations of Lie algebras
Abstract
In this paper we study integrability and algebraic integrability properties of certain matrix Schrödinger 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 CalogeroSutherland 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 CalogeroSutherland 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 CalogeroSutherland operators.
 Publication:

arXiv eprints
 Pub Date:
 March 1994
 arXiv:
 arXiv:hepth/9403135
 Bibcode:
 1994hep.th....3135E
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Quantum Algebra
 EPrint:
 19 pages