Perturbative breaking of the pseudospin symmetry in the relativistic harmonic oscillator

We show that relativistic mean fields theories with scalar, $S$, and vector, $V$, quadratic radial potentials can generate a harmonic oscillator with exact pseudospin symmetry {\it and positive energy bound states} when $S=-V$. The eigenenergies are quite different from those of the non-relativistic harmonic oscillator. We also discuss a mechanism for perturbatively breaking this symmetry by introducing a tensor potential. Our results shed light into the intrinsic relativistic nature of the pseudospin symmetry, which might be important in high density systems such as neutron stars.