On the behaviour of the two-dimensional Hamiltonian $-\Delta +\lambda \left[\delta \right(\overrightarrow{x}+{\overrightarrow{x}}_0)+\delta (\overrightarrow{x}-{\overrightarrow{x}}_0\left)\right]$ as the distance between the two centres vanishes

In this note we continue our analysis of the behaviour of self-adjoint Hamiltonians with a pair of identical point interactions symmetrically situated around the origin perturbing various types of 'free Hamiltonians' as the distance between the two centres shrinks to zero. In particular, by making the coupling constant to be renormalised dependent also on the separation distance between the centres of the two point interactions, we prove that also in two dimensions it is possible to define the unique self-adjoint Hamiltonian that, differently from the one studied in detail in Albeverio's monograph on point interactions, behaves smoothly as the separation distance vanishes. In fact, we rigorously prove that such a two-dimensional Hamiltonian converges in the norm resolvent sense to the one of the negative two-dimensional Laplacian perturbed by a single attractive point interaction situated at the origin having double strength, thus making this two-dimensional model similar to its one-dimensional analogue (not requiring the renormalisation procedure). (paper)