Unstable quantum oscillator with point interactions: Maverick resonances, antibound states and other surprises

In the search for solvable or quasi-solvable models for resonances, we consider a one-dimensional potential, which is a harmonic oscillator for x<0, has a point potential at the origin of the form aδ(x)+bδ′(x) and no interaction for x>0. After a study of this model, we add a mass jump at the origin and study the effect of the combination of the mass jump and the point potential. We obtain the behavior of resonances, bound and antibound states in terms of given parameters. In spite of the simplicity of the model, it shows quite interesting and unexpected features.