The Hamiltonian of the harmonic oscillator with an attractive δ'- interaction centred at the origin as approximated by the one with a triple of attractive δ-interactions

In this note we provide an alternative way of defining the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive delta- interaction,of strength beta, centred at 0 (the bottom of the confining parabolic potential), that was rigorously defined in a previous paper by means of a ‘coupling constant renormalisation’. Here we get the Hamiltonian as a norm resolvent limit of the harmonic oscillator Hamiltonian perturbed by a triple of attractive delta-interactions, thus extending the Cheon–Shigehara approximation to the case in which a confining harmonic potential is present.