Algebraic study on the $A_{N-1}$- and $B_N$-Calogero models with bosonic, fermionic and distinguishable particles

TL;DR

This paper uses algebraic methods with Dunkl--Cherednik operators to analyze multivariable Hermite and Laguerre polynomials related to Calogero models, providing explicit formulas for eigenstates and their norms across different particle types.

Contribution

It introduces algebraic Rodrigues formulas for non-symmetric polynomials and unified norm calculations for all particle cases in Calogero models.

Findings

Explicit Rodrigues formulas for non-symmetric polynomials

Unified algebraic calculation of eigenstate norms

Construction of symmetric and anti-symmetric eigenstates

Abstract

Through an algebraic method using the Dunkl--Cherednik operators, the multivariable Hermite and Laguerre polynomials associated with the $A_{N-1}$- and $B_N$-Calogero models with bosonic, fermionic and distinguishable particles are investigated. The Rodrigues formulas of column type that algebraically generate the monic non-symmetric multivariable Hermite and Laguerre polynomials corresponding to the distinguishable case are presented. Symmetric and anti-symmetric polynomials that respectively give the eigenstates for bosonic and fermionic particles are also presented by the symmetrization and anti-symmetrization of the non-symmetric ones. The norms of all the eigenstates for all cases are algebraically calculated in a unified way.