In this note we provide an alternative way of defining the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive - interaction, of strength , 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 -interactions, thus extending the Cheon–Shigehara approximation to the case in which a confining harmonic potential is present.

We study three solvable two-dimensional systems perturbed by a point interaction centered at the origin. The unperturbed systems are the isotropic harmonic oscillator, a square pyramidal potential and a combination thereof. We study the spectrum of the perturbed systems. We show that, while most eigenvalues are not affected by the point perturbation, a few of them are strongly perturbed. We show that for some values of one parameter, these perturbed eigenvalues may take lower values than the immediately lower eigenvalue, so that level crossings occur. These level crossings are studied in some detail.