The energy level structure of a variety of one-dimensional confining potentials and the effects of a local singular perturbation

Motivated by current interest in quantum confinement potentials, especially with respect to the Stark spectroscopy of new types of quantum wells, we examine several novel one-dimensional singular oscillators. A Green function method is applied, the construction of the necessary resolvents is reviewed and several new ones are introduced. In addition, previous work on the singular harmonic oscillator model, introduced by Avakian et al. (Phys. Lett. A, 124, 233 (1987). doi:10.1016/0375-9601(87)90627-X0) is reproduced to verify the method and results. A novel feature is the determination of the spectra of asymmetric hybrid linear and quadratic potentials. As in previous work, the singular perturbations are modeled by delta functions.