Non-real eigenvalues of the Harmonic Oscillator perturbed by an odd, two-point $\delta$-potential

In this paper, we consider the perturbations of the Harmonic Oscillator Operator by an odd pair of point interactions: $z (\delta(x - b) - \delta(x + b))$. We study the spectrum by analyzing a convenient formula for the eigenvalue. We conclude that if $z=ir$, $r$ real, as $r\to \mathrm{\infty }$, the number of non-real eigenvalues tends to infinity.