Symmetric Fock space and orthogonal symmetric polynomials associated with the Calogero model

TL;DR

This paper constructs a new orthogonal basis for the Calogero model's Hilbert space using a similarity transformation, revealing two inequivalent sets of conserved operators and contrasting with the known Jack polynomial-based basis.

Contribution

It introduces a novel orthogonal basis for the Calogero model via a similarity transformation, showing the existence of two algebraically inequivalent sets of conserved operators.

Findings

New orthogonal basis different from Jack polynomial variants

Existence of two inequivalent sets of conserved operators

Confirmed similar structure for B_N-Calogero model

Abstract

Using a similarity transformation that maps the Calogero model into $N$ decoupled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their simultaneous eigenfunctions. The simultaneous eigenfunction is a deformation of the symmetrized number state (bosonic state) and forms an orthogonal basis of the Hilbert (Fock) space of the model. This orthogonal basis is different from the known one that is a variant of the Jack polynomial, i.e., the Hi-Jack polynomial. This fact shows that the conserved operators derived by the similarity transformation and those derived by the Dunkl operator formulation do not commute. Thus we conclude that the Calogero model has two, algebraically inequivalent sets of mutually commuting conserved operators, as is the case with the hydrogen atom. We also confirm the same story for the $B_{N}$-Calogero model.