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.
Categories: quant-ph hep-th