Generalized oscillator representations for Calogero Hamiltonians

TL;DR

This paper extends oscillator representations to all Calogero Hamiltonians with coupling constant , providing a comprehensive framework that includes highly nonunique cases but identifies an optimal representation for each.

Contribution

It introduces generalized oscillator representations for all Calogero Hamiltonians with , broadening previous work limited to nonnegative cases.

Findings

Constructed generalized oscillator representations for all  Calogero Hamiltonians.

Identified an optimal oscillator representation for each Hamiltonian.

Showed that these representations are generally highly nonunique.

Abstract

This paper is a natural continuation of the previous paper J.Phys. A: Math.Theor. 44 (2011) 425204, arXiv 0907.1736 [quant-ph] where oscillator representations for nonnegative Calogero Hamiltonians with coupling constant $\alpha\geq-1/4$ were constructed. Here, we present generalized oscillator representations for all Calogero Hamiltonians with $\alpha\geq-1/4$.These representations are generally highly nonunique, but there exists an optimum representation for each Hamiltonian.