A one dimensional model showing a quantum phase transition based on a singular potential

We study a one-dimensional singular potential plus three types of regular interactions: constant electric field, harmonic oscillator and infinite square well. We use the Lippman-Schwinger Green function technique in order to search for the bound state energies. In the electric field case the unique bound state coincides with that found in an earlier study as the field is switched off. For non-zero field the ground state is shifted and positive energy "quasibound states" appear. For the harmonic oscillator we find a quantum phase transition of a novel type. This behavior does not occur in the corresponding case of an infinite square well and demonstrates the influence of quantum non-locality.