In these lectures, I review some recent results on the Calogero-Sutherland model and the Haldane Shatry-chain. The list of topics I cover are the following: 1) The Calogero-Sutherland Hamiltonien and fractional stastistics. The form factor of the density operator. 2) The Dunkl operators and their relations with monodromy matrices, Yangians and aane-Hecke algebras. 3) The Haldane-Shastry chain i...

It is known that the trigonometric Calogero-Sutherland model is obtained by the trigonometric limit (τ → √ −1∞) of the elliptic CalogeroMoser model, where (1, τ) is a basic period of the elliptic function. We show that for all square-integrable eigenstates and eigenvalues of the Hamiltonian of the Calogero-Sutherland model, if exp(2π √ −1τ) is small enough then there exist square-integrable eig...

Within the supersymmetric quantum mechanics the necessary regularization of the poles of the superpotentials on the real line of coordinates x may be most easily mediated by a small constant shift of this axis into complex plane. Detailed attention is paid here to the resulting recipe which works, in effect, with non-Hermitian (a. k. a. PT −symmetric or pseudo-Hermitian) Hamiltonians. Besides a...

A family of systems related to a linear and bilinear evolution of roots of polynomials in the complex plane is introduced. Restricted to the line, the evolution induces dynamics of the Coulomb charges (or point vortices) in external potentials, while its fixed points correspond to equilibriums of charges in the plane. The construction reveals a direct connection with the theories of the Caloger...

We use the Thomas-Fermi method to examine the thermodynamics of particles obeying Haldane exclusion statistics. Specifically, we study Calogero-Sutherland particles placed in a given external potential in one dimension. For the case of a simple harmonic potential (constant density of states), we obtain the exact one-particle spatial density and a closed form for the equation of state at finite ...

The original Calogero model describes N quantum particles on the line interacting pairwise through a inverse square potential. It is well known to be integrable and solvable. Here we extend the Calogero Hamiltonian by means of new interactions which are PT-symmetric but not self adjoint. Some of these new interactions lead to integrable PT-symmetric Hamiltonians; the algebraic properties furthe...

We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These are obtained by a spin-pole ansatz including dynamical background term; we show that this solves sncILW provided spins and poles satisfy $A$-type Calogero-Moser (sCM) system certain constraints on initial The key to result is B\"{a}cklund transfo...

A novel realization is provided for the scattering states of the N -particle Calogero-Moser Hamiltonian. They are explicitly shown to be the coherent states of the singular oscillators of the Calogero-Sutherland model. Our algebraic treatment is straightforwardly extendable to a large number of few and many-body interacting systems in one and higher dimensions. PACS: 03.65.-w, 03.65.Ge Typeset ...