Large Oscillator representations for self-adjoint Calogero Hamiltonians

Abstract

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.