Rodrigues Formula for Hi-Jack Symmetric Polynomials Associated with the Quantum Calogero Model

Abstract

The Hi-Jack symmetric polynomials, which are associated with the simultaneous eigenstates for the first and second conserved operators of the quantum Calogero model, are studied. Using the algebraic properties of the Dunkl operators for the model, we derive the Rodrigues formula for the Hi-Jack symmetric polynomials. Some properties of the Hi-Jack polynomials and the relationships with the Jack symmetric polynomials and with the basis given by the QISM approach are presented. The Hi-Jack symmetric polynomials are strong candidates for the orthogonal basis of the quantum Calogero model.KEYWORDS: quantum Calogero model, Hi-Jack symmetric polynomials, Rodrigues formula, quantum inverse scattering method (QISM), Dunkl operator §1. IntroductionExact solutions for the Schrödinger equations have provided important significance in physics and mathematical physics. Most of us have studied the Laguerre polynomials and the spherical harmonics in the theory of the hydrogen atom, and the Hermite polynomials and their Rodrigues formula in the theory of the quantum harmonic oscillator. The former is also a good example that shows the role of conserved operators in quantum mechanics. The hydrogen atom has three, independent and mutually commuting conserved operators, namely, the Hamiltonian, the total angular momentum and its z-axis component. The simultaneous eigenstates for the three conserved operators give the orthogonal basis of the hydrogen atom. A classical system with a set of independent and mutually Poisson commuting (involutive) conserved quantities whose number of elements is the same as the degrees of freedom of the system can be integrated by quadrature. This is guaranteed by the Liouville theorem. Such a system is called the completely integrable system. Quantum systems with enough number of such conserved operators are analogously called quantum integrable systems. The hydrogen atom is a simple example of the quantum integrable system. Among the various quantum integrable systems, onedimensional quantum many-body systems with inversesquare long-range interactions are now attracting much interests of theoretical physicists. Of the various integrable inverse-square-interaction models, the quantum Calogero model 1) has the longest history. Its Hamilto- *

Supersymmetric Calogero–Moser–Sutherland models and Jack superpolynomials

“…Some of these results -in particular, the creation-operator formalism -have been extended to the construction of the hi-Jack polynomials [38], which are eigenfunctions of the CMS model with an inverse-square interaction augmented by an harmonic confining term, and of the Macdonald polynomials [39,40], which are eigenfunctions of the trigonometric Ruijsenaars-Schneider model [41], a relativistic version of the tCMS model. In the latter case, integral formulas have also been obtained (see e.g., [42] …”

Orthogonality of the Hi-Jack Polynomials Associated with the Calogero Model

“…Thus we call the unidentified simultaneous eigenfunctions of the Calogero model Hi-Jack (hidden-Jack) polynomials [25][26][27]. We shall extend the method Lapointe and Vinet developed to construct the Rodrigues formula for the Jack polynomials [18,19] to the quantum Calogero model and derive the Rodrigues formula for the Hi-Jack polynomials [25,26]. We shall study some properties of the Hi-Jack polynomials such as integrality, triangularity and orthogonality.…”

Orthogonal basis for the energy eigenfunctions of the Chern-Simons matrix model

“…Comparing (37) to the orthogonal basis of the energy eigenfunctions of the Calogero model [12]- [17], we conclude that the polynomials J {λ} ought to be the symmetric Jack polynomials. The inhomogeneous polynomials e −Ô L 4B J {λ} ( √ 2Bx i ) are the symmetric Hi-Jack polynomials [14] - [17] which provide an orthogonal basis for the Calogero model with the integration measure [15]- [16] ∆ 2l+2 e −B x 2…”