Pseudo-bosons for the D2 type quantum Calogero model

In the first part of this paper we show how a simple system, a 2-dimensional quantum harmonic oscillator, can be described in terms of pseudo-bosonic variables. This apparently strange choice is useful when the natural Hilbert space of the system, L2(R2) in this case, is, for some reason, not the most appropriate. This is exactly what happens for the D2 type quantum Calogero model considered in the second part of the paper, where the Hilbert space L2(R2) appears to be an unappropriate choice, since the eigenvectors of the relevant hamiltonian are not square-integrable. Then we discuss how a certain intertwining operator arising from the model can be used to fix a different Hilbert space more useful.