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

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.