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# Imprimitively generated Lie-algebraic Hamiltonians and separation of variables

Published:1998-12-01
Printed: Dec 1998
• Robert Milson
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## Abstract

Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose domain is a subset of the Euclidean plane admits a separation of variables. A proof of this conjecture is given in those cases where the generating Lie-algebra acts imprimitively. The general form of the conjecture is false. A counter-example is given based on the trigonometric Olshanetsky-Perelomov potential corresponding to the $A_2$ root system.
 MSC Classifications: 35Q40 - PDEs in connection with quantum mechanics 53C30 - Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 81R05 - Finite-dimensional groups and algebras motivated by physics and their representations [See also 20C35, 22E70]