Quantum constants of the motion for two-dimensional systems E. G. Kalnins Department of Mathematics, University of Waikato, Hamilton, New Zealand, e.kalnins@waikato.ac.nz W. Miller, Jr. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, U.S.A., miller@ima.umn.edu and G. Pogosyan Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, 14980, Russia, pogosyan@thsun1.jinr.dubna.su March 5, 2003 Abstract Consider a non-relativistic Hamiltonian operator H in 2 dimen- sions consisting of a kinetic energy term plus a potential. We show that if the associated Schr¨odinger eigenvalue equation admits an orthogonal separation of variables then it is possible to generate algorithmically a canonical basis Q, P where P 1 = H , P 2 , are the other 2nd-order con- stants of the motion associated with the separable coordinates, and [Q i ,Q j ]=[P i ,P j ] = 0, [Q i ,P j ]= δ ij . The 3 operators Q 2 ,P 1 ,P 2 form a basis for the invariants. In general these are infinite-order differen- tial operators. We shed some light on the general question of exactly when the Hamiltonian admits a constant of the motion that is poly- nomial in the momenta. We go further and consider all cases where the Hamilton-Jacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordi- nates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion. 1