Pseudo-bosons for the 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, L 2 ( R 2 ) in this case, is, for some reason, not the most appropriate. This is exactly what happens for the D 2 type quantum Calogero model considered in the second part of the paper, where the Hilbert space L 2 ( R 2 ) 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.