We investigate the dynamics of the spin-less relativistic particle subject to an external field of a harmonic oscillator potential. The Klein-Gordon equation with one- and three-dimensional vector and scalar parabolic potentials is solved using the expansion of the wavefunction in properly selected basis-sets. The resonance states are determined using the complex coordinate rotation method. The analytic expressions for the first- and second-order relativistic energy corrections are derived perturbatively. The relativistic model of the harmonic oscillator in the momentum representation, originally proposed by Znojil, is also discussed.
The classification of states based on good quantum numbers for the two-dimensional Coulomb problem is proposed. The first order magnetic energy corrections are calculated using exact field-free analytic solutions of the Dirac equation as a zero-order approximation.
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