  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.
