Related papers: Some Properties of the Calogero-Sutherland Model w…

Using a collective field method, we obtain explicit solutions of the generalized Calogero-Sutherland models that are characterized by the roots of the classical groups $B_N$ and $C_N$. Starting from the explicit wave functions for $A_{N-1}$…

The Hamiltonian of the quantum Calogero-Sutherland model of $N$ identical particles on the circle with $1/r^{2}$ interactions has eigenfunctions consisting of Jack polynomials times the base state. By use of the generalized Jack polynomials…

In a previous paper we introduced and developed a recursive construction of joint eigenfunctions $J_N(a_+,a_-,b;x,y)$ for the Hamiltonians of the hyperbolic relativistic Calogero-Moser system with arbitrary particle number $N$. In this…

Using the ideas of supersymmetry and shape invariance we re-derive the spectrum of the $A_{N-1}$ and $BC_N$ Calogero-Sutherland model. We briefly discuss as to how to obtain the corresponding eigenfunctions. We also discuss the difficulties…

Calogero-Sutherland models of type $BC_N$ are known to be relevant to the physics of one-dimensional quantum impurity effects. Here we represent certain correlation functions of these models in terms of generalized hypergeometric functions.…

In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions $J_N(a_+, a_-,b;x,y)$ of the Hamiltonians arising in the integrable $N$-particle systems of hyperbolic…

The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group $B_2$, which is the symmetry group of the square. The angular momentum operator is also modified with…

We propose to parametrize the configuration space of one-dimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the…

We express the Hamiltonian of the quantum trigonometric Calogero-Sutherland model related to the Lie algebra $D_4$ in terms of a set of Weyl-invariant variables, namely, the characters of the fundamental representations of the Lie algebra.…

We show that Calogero-Sutherland models for interacting particles have a natural supersymmetric extension. For the construction, we use Jacobians which appear in certain superspaces. Some of the resulting Hamiltonians have a direct physics…

The BC-type Calogero-Sutherland model (CSM) is an integrable extension of the ordinary A-type CSM that possesses a reflection symmetry point. The BC-CSM is related to the chiral classes of random matrix ensembles (RMEs) in exactly the same…

A quantum integrable model is considered which describes a quantization of affine hyper-elliptic Jacobian. This model is shown to possess the property of duality: a dual model with inverse Planck constant exists such that the…

The $B_N$-type Calogero-Sutherland-Moser system in one-dimension is shown to be equivalent to a set of decoupled oscillators by a similarity transformation. This result is used to show the connection of the $A_N$ and $B_N$ type models and…

We disclose the Lie algebraic structure of two extended Sutherland models. Their Hamiltonians are BC_N, and A_N Sutherland Hamiltonians with some additional terms. We show that both Hamiltonians can be written in the quadratic forms of…

We construct certain eigenfunctions of the Calogero-Sutherland hamiltonian for particles on a circle, with mixed boundary conditions. That is, the behavior of the eigenfunction, as neighbouring particles collide, depend on the pair of…

In a recent work, we have initiated the theory of N=2 symmetric superpolynomials. As far as the classical bases are concerned, this is a rather straightforward generalization of the N=1 case. However this construction could not be…

We review a recent construction of an explicit analytic series representation for symmetric polynomials which up to a groundstate factor are eigenfunctions of Calogero-Sutherland type models. We also indicate a generalisation of this result…

Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they…

We present how explicit eigenfunctions of the principal Hamiltonian for the $BC_{m}$ relativistic Calogero-Moser-Sutherland model, due to van Diejen, can be constructed using gauge and integral transformations. In particular, we find that…

We obtain symmetric joint eigenfunctions for the commuting PDOs associated to the hyperbolic Calogero-Moser N-particle system. The eigenfunctions are constructed via a recursion scheme, which leads to representations by multidimensional…