Bound states of relativistic spinless particles in a mix of circularly symmetric vector and scalar harmonic oscillators

Abstract We study the dynamics of relativistic spinless particles moving in a plane when there is circular symmetry. The general formalism for solving the Klein–Gordon equation in cylindrical coordinates for such systems is presented, as well as the conserved observables and the corresponding quantum numbers. We look for bound solutions of the corresponding Klein–Gordon equation when one has vector and scalar circularly symmetric harmonic oscillator potentials. Both positive and negative bound solutions are considered when there is either equal vector and scalar potentials or symmetric vector and scalar potentials, and it is shown how both cases are related through charge conjugation. We compute the non-relativistic limit for those cases, and show that for symmetric scalar and vector potentials the limit does not exist in the first order of an harmonic oscillator frequency, recovering a known result from the Dirac equation with the same kind of potentials.