Large Oscillator representations for self-adjoint Calogero Hamiltonians

TL;DR

This paper explores the oscillator representations of self-adjoint Calogero Hamiltonians, identifying those that can be expressed as oscillator type operators and deriving the associated elementary operators.

Contribution

It introduces oscillator representations for Calogero Hamiltonians and characterizes which self-adjoint operators admit such representations.

Findings

Some self-adjoint Calogero Hamiltonians are of oscillator type.

Explicit elementary operators corresponding to these Hamiltonians are derived.

The properties of these oscillator representations are analyzed.

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.