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

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.Comment: 17 pages, LaTeX file using jpsj.sty (ver. 0.8), cite.sty, subeqna.sty, subeqn.sty, jpsjbs1.sty and jpsjbs2.sty (all included.) You can get all the macros from ftp.u-tokyo.ac.jp/pub/SOCIETY/JPSJ

Rodrigues Formula for the Nonsymmetric Multivariable Hermite Polynomial

Applying a method developed by Takamura and Takano for the nonsymmetric Jack polynomial, we present the Rodrigues formula for the nonsymmetric multivariable Hermite polynomial.Comment: 5 pages, LaTe

Rodrigues Formula for the Nonsymmetric Multivariable Laguerre Polynomial

Extending a method developed by Takamura and Takano, we present the Rodrigues formula for the nonsymmetric multivariable Laguerre polynomials which form the orthogonal basis for the $B_{N}$-type Calogero model with distinguishable particles. Our construction makes it possible for the first time to algebraically generate all the nonsymmetric multivariable Laguerre polynomials with different parities for each variable.Comment: 6 pages, LaTe

Equivalence of the Calogero-Sutherland Model to Free Harmonic Oscillators

A similarity transformation is constructed through which a system of particles interacting with inverse-square two-body and harmonic potentials in one dimension, can be mapped identically, to a set of free harmonic oscillators. This equivalence provides a straightforward method to find the complete set of eigenfunctions, the exact constants of motion and a linear $W_{1+\infty}$ algebra associated with this model. It is also demonstrated that a large class of models with long-range interactions, both in one and higher dimensions can be made equivalent to decoupled oscillators.Comment: 9 pages, REVTeX, Completely revised, few new equations and references are adde

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

We study the spectrum of the Chern-Simons matrix model and identify an orthogonal set of states. The connection to the spectrum of the Calogero model is discussed.Comment: 11 pages, LaTeX, minor typo corrections, section 6 slightly extended to include more information on Jack polynomial

Exact spectrum and partition function of SU(m|n) supersymmetric Polychronakos model

By using the fact that Polychronakos-like models can be obtained through the `freezing limit' of related spin Calogero models, we calculate the exact spectrum as well as partition function of SU(m|n) supersymmetric Polychronakos (SP) model. It turns out that, similar to the non-supersymmetric case, the spectrum of SU(m|n) SP model is also equally spaced. However, the degeneracy factors of corresponding energy levels crucially depend on the values of bosonic degrees of freedom (m) and fermionic degrees of freedom (n). As a result, the partition functions of SP models are expressed through some novel q-polynomials. Finally, by interchanging the bosonic and fermionic degrees of freedom, we obtain a duality relation among the partition functions of SP models.Comment: Latex, 20 pages, no figures, minor typos correcte

Exact solution of Calogero model with competing long-range interactions

An integrable extension of the Calogero model is proposed to study the competing effect of momentum dependent long-range interaction over the original {1 \ov r^2} interaction. The eigenvalue problem is exactly solved and the consequences on the generalized exclusion statistics, which appears to differ from the exchange statistics, are analyzed. Family of dual models with different coupling constants is shown to exist with same exclusion statistics.Comment: Revtex, 6 pages, 1 figure, hermitian variant of the model included, final version to appear in Phys. Rev.

Super-Calogero-Moser-Sutherland systems and free super-oscillators : a mapping

We show that the supersymmetric rational Calogero-Moser-Sutherland (CMS) model of A_{N+1}-type is equivalent to a set of free super-oscillators, through a similarity transformation. We prescribe methods to construct the complete eigen-spectrum and the associated eigen-functions, both in supersymmetry preserving as well as supersymmetry breaking phases, from the free super-oscillator basis. Further we show that a wide class of super-Hamiltonians realizing dynamical OSp(2|2) supersymmetry, which also includes all types of rational super-CMS as a small subset, are equivalent to free super-oscillators. We study BC_{N+1}-type super-CMS model in some detail to understand the subtleties involved in this method.Comment: 27 pages, RevTeX, No figures; Minor clarifications added, version to appear in Nuclear Physics

Quantum Calogero-Moser Models: Integrability for all Root Systems

The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the (A) series, i.e. (su(N)) type, root systems.Comment: 45 pages, LaTeX2e, no figure

Renormalized Harmonic-Oscillator Description of Confined Electron Systems with Inverse-Square Interaction

An integrable model for SU($\nu$) electrons with inverse-square interaction is studied for the system with confining harmonic potential. We develop a new description of the spectrum based on the {\it renormalized harmonic-oscillators} which incorporate interaction effects via the repulsion of energy levels. This approach enables a systematic treatment of the excitation spectrum as well as the ground-state quantities.Comment: RevTex, 7 page