On uniqueness of commutative rings of Weyl group invariant differential operators (Q1364333)

The construction and the classification of the commutative rings of differential operators which contain the Laplacian \(H={1\over 2}\sum^n_{i=1} {\partial^2 \over \partial x_i^2} +R(x)\) is presented. The differential operators \(P\) commuting with \(H\) are determined without assuming that \(P\) is \(W\)-invariant or that it has a constant symbol. The uniqueness of commutative rings of Weyl group invariant differential operators is proven for trigonometric or elliptic potentials. Counterexamples are constructed for the rational potential cases.