Pseudo-bosons for the $D_2$ type quantum Calogero model

TL;DR

This paper demonstrates how pseudo-bosonic variables can be used to analyze the $D_2$ quantum Calogero model, addressing issues with the natural Hilbert space and proposing a modified space via an intertwining operator.

Contribution

It introduces the use of pseudo-bosonic variables for the $D_2$ Calogero model and shows how to select a more appropriate Hilbert space using an intertwining operator.

Findings

Pseudo-bosonic variables effectively describe the 2D quantum harmonic oscillator.

The natural Hilbert space $L^2({\bf R}^2)$ is unsuitable for the $D_2$ Calogero model.

An intertwining operator can be used to define a more appropriate Hilbert space.

Abstract

In the first part of this paper we show how a simple system, a 2-dimensional quantum harmonic oscillator, can be described in terms of pseudo-bosonic variables. This apparently {\em strange} choice is useful when the {\em natural} Hilbert space of the system, $L^2({\bf R}^2)$ in this case, is, for some reason, not the most appropriate. This is exactly what happens for the $D_2$ type quantum Calogero model considered in the second part of the paper, where the Hilbert space $L^2({\bf R}^2)$ appears to be an unappropriate choice, since the eigenvectors of the relevant hamiltonian are not square-integrable. Then we discuss how a certain intertwining operator arising from the model can be used to fix a different Hilbert space more {\em useful}.