Polynomial Solutions of Generalized Quartic Anharmonic Oscillators

We introduce a nilpotent group to write a generalized quartic anharmonic oscillator Hamiltonian as a polynomial in the generators of the group. Energy eigenvalues are then seen to depend on the values of the two Casimir operators of the group. This dependence exhibits a scaling law which follows from the scaling properties of the group generators. Demanding that the potential give rise to polynomial solutions in a particular Lie algebra element puts constraints on the four potential parameters, leaving only two of them free. For potentials satisfying such constraints at least one of the energy eigenvalues and the corresponding eigenfunctions can be obtained in closed analytic form; examples, beyond those available in the literature, are given. Finally, we find solutions for particles in external electromagnetic fields in terms of these quartic polynomial solutions.