  In the article arXiv:0903.5277 [quant-ph], we have presented a mathematically
rigorous quantum-mechanical treatment of a one-dimensional motion of a particle
in the Calogero potential $V(x)=\alpha x^{-2}$. In such a way, we have
described all possible s.a. operators (s.a. Hamiltonians) associated with the
formal differential expression $\check{H}=-d_{x}^{2}+\alpha x^{-2}$ for the
Calogero Hamiltonian. Here, we discuss a new aspect of the problem, the
so-called oscillator representation for the Calogero Hamiltonians. As it is
know, operators of the form $\hat{N}=\hat{a}^{+}\hat{a}$ and
$\hat{A}=\hat{a}\hat{a}^{+}$ are called operators of oscillator type.
Oscillator type operators obey several useful properties in case if the
elementary operator $\hat{a}$ and $\hat{a}^{+}$ are densely defined. It turns
out that some s.a. Calogero Hamiltonians are oscillator type operators. We
describe such Hamiltonians and find the corresponding mutually adjoint
elementary operators.
