Imprimitively generated Lie-algebraic Hamiltonians and separation of variables
Printed: Dec 1998
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.
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]