Common Algebraic Structure for the Calogero-Sutherland Models

We investigate common algebraic structure for the rational and trigonometric Calogero-Sutherland models by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis for the rational case.Comment: 7 pages, LaTeX, no figures, some text and references added, minor misprints correcte

The Calogero-Moser equation system and the ensemble average in the Gaussian ensembles

From random matrix theory it is known that for special values of the coupling constant the Calogero-Moser (CM) equation system is nothing but the radial part of a generalized harmonic oscillator Schroedinger equation. This allows an immediate construction of the solutions by means of a Rodriguez relation. The results are easily generalized to arbitrary values of the coupling constant. By this the CM equations become nearly trivial. As an application an expansion for in terms of eigenfunctions of the CM equation system is obtained, where X and Y are matrices taken from one of the Gaussian ensembles, and the brackets denote an average over the angular variables.Comment: accepted by J. Phys.

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

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

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.

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

Exclusion statistics,operator algebras and Fock space representations

We study exclusion statistics within the second quantized approach. We consider operator algebras with positive definite Fock space and restrict them in a such a way that certain state vectors in Fock space are forbidden ab initio.We describe three characteristic examples of such exclusion, namely exclusion on the base space which is characterized by states with specific constraint on quantum numbers belonging to base space M (e.g. Calogero-Sutherland type of exclusion statistics), exclusion in the single-oscillator Fock space, where some states in single oscillator Fock space are forbidden (e.g. the Gentile realization of exclusion statistics) and a combination of these two exclusions (e.g. Green's realization of para-Fermi statistics). For these types of exclusions we discuss extended Haldane statistics parameters g, recently introduced by two of us in Mod.Phys.Lett.A 11, 3081 (1996), and associated counting rules. Within these three types of exclusions in Fock space the original Haldane exclusion statistics cannot be realized.Comment: Latex,31 pages,no figures,to appear in J.Phys.A : Math.Ge

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

Integrable 1/r21/r^2 Spin Chain with Reflecting End

A new integrable spin chain of the Haldane-Shastry type is introduced. It is interpreted as the inverse-square interacting spin chain with a {\it reflecting end}. The lattice points of this model consist of the square roots of the zeros of the Laguerre polynomial. Using the ``exchange operator formalism'', the integrals of motion for the model are explicitly constructed.Comment: 13 pages, REVTeX3, with minor correction

A Unified Algebraic Approach to Few and Many-Body Correlated Systems

The present article is an extended version of the paper {\it Phys. Rev.} {\bf B 59}, R2490 (1999), where, we have established the equivalence of the Calogero-Sutherland model to decoupled oscillators. Here, we first employ the same approach for finding the eigenstates of a large class of Hamiltonians, dealing with correlated systems. A number of few and many-body interacting models are studied and the relationship between their respective Hilbert spaces, with that of oscillators, is found. This connection is then used to obtain the spectrum generating algebras for these systems and make an algebraic statement about correlated systems. The procedure to generate new solvable interacting models is outlined. We then point out the inadequacies of the present technique and make use of a novel method for solving linear differential equations to diagonalize the Sutherland model and establish a precise connection between this correlated system's wave functions, with those of the free particles on a circle. In the process, we obtain a new expression for the Jack polynomials. In two dimensions, we analyze the Hamiltonian having Laughlin wave function as the ground-state and point out the natural emergence of the underlying linear $W_{1+\infty}$ symmetry in this approach.Comment: 18 pages, Revtex format, To appear in Physical Review