Third-order constants of motion in quantum mechanics

We investigate the general form of a third-order linear differential operator that is required to commute with the Schrödinger Hamiltonian in two dimensions, and find that the third-order part must be a polynomial of third degree in the generators of the Euclidean group. Partial differential equations that the potentialV must satisfy are derived, and solved for the special cases where the Schrödinger equation separates in polar or Cartesian coordinates. The functionsV thus obtained are nonsingular, but are periodic through elliptic functions. After separation of variables, the Schrödinger equation gives Lame's equation.