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

TL;DR

This paper investigates the energy levels of various one-dimensional confining potentials, including singular and asymmetric types, using Green function methods and delta function perturbations, relevant for quantum well spectroscopy.

Contribution

It introduces new resolvent constructions for singular oscillators and determines spectra of asymmetric hybrid potentials, extending previous models.

Findings

Spectra of asymmetric hybrid linear and quadratic potentials are determined.

Green function method effectively analyzes singular perturbations.

Reproduction of previous results verifies the new approach.

Abstract

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. is reproduced to verify the method and results. A novel features 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.