Large Oscillator representations for self-adjoint Calogero Hamiltonians

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 $\widehat{N}={\widehat{a}}^{+}\widehat{a}$ and $\widehat{A}=\widehat{a}{\widehat{a}}^{+}$ are called operators of oscillator type. Oscillator type operators obey several useful properties in case if the elementary operator $\widehat{a}$ and ${\widehat{a}}^{+}$ 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.