Knizhnik-Zamolodchikov equations and the Calogero-Sutherland-Moser integrable models with exchange terms

It is shown that from some solutions of generalized Knizhnik-Zamolodchikov equations one can construct eigenfunctions of the Calogero-Sutherland-Moser Hamiltonians with exchange terms, which are characterized by any given permutational symmetry under particle exchange. This generalizes some results previously derived by Matsuo and Cherednik for the ordinary Calogero-Sutherland-Moser Hamiltonians.Comment: 13 pages, LaTeX, no figures, to be published in J. Phys.

Solitons in the Calogero-Sutherland Collective-Field Model

In the Bogomol'nyi limit of the Calogero-Sutherland collective-field model we find static-soliton solutions. The solutions of the equations of motion are moving solitons, having no static limit for \l>1. They describe holes and lumps, depending on the value of the statistical parametar \l.Comment: minor correction

Poisson Structures for Aristotelian Model of Three Body Motion

We present explicitly Poisson structures, for both time-dependent and time-independent Hamiltonians, of a dynamical system with three degrees of freedom introduced and studied by Calogero et al [2005]. For the time-independent case, new constant of motion includes all parameters of the system. This extends the result of Calogero et al [2009] for semi-symmetrical motion. We also discuss the case of three bodies two of which are not interacting with each other but are coupled with the interaction of third one

Upper and lower limits on the number of bound states in a central potential

In a recent paper new upper and lower limits were given, in the context of the Schr\"{o}dinger or Klein-Gordon equations, for the number $N_{0}$ of S-wave bound states possessed by a monotonically nondecreasing central potential vanishing at infinity. In this paper these results are extended to the number $N_{\ell}$ of bound states for the $\ell$-th partial wave, and results are also obtained for potentials that are not monotonic and even somewhere positive. New results are also obtained for the case treated previously, including the remarkably neat \textit{lower} limit $N_{\ell}\geq \{\{[\sigma /(2\ell+1)+1]/2\}\}$ with $V(r)| ^{1/2}]$ (valid in the Schr\"{o}dinger case, for a class of potentials that includes the monotonically nondecreasing ones), entailing the following \textit{lower} limit for the total number $N$ of bound states possessed by a monotonically nondecreasing central potential vanishing at infinity: N\geq \{\{(\sigma+1)/2\}\} {(\sigma+3)/2\} \}/2 (here the double braces denote of course the integer part).Comment: 44 pages, 5 figure

Sufficient conditions for the existence of bound states in a central potential

We show how a large class of sufficient conditions for the existence of bound states, in non-positive central potentials, can be constructed. These sufficient conditions yield upper limits on the critical value, $g_{\rm{c}}^{(\ell)}$, of the coupling constant (strength), $g$, of the potential, $V(r)=-g v(r)$, for which a first $\ell$-wave bound state appears. These upper limits are significantly more stringent than hitherto known results.Comment: 7 page

Spectrum of a spin chain with inverse square exchange

The spectrum of a one-dimensional chain of $SU(n)$ spins positioned at the static equilibrium positions of the particles in a corresponding classical Calogero system with an exchange interaction inversely proportional to the square of their distance is studied. As in the translationally invariant Haldane--Shastry model the spectrum is found to exhibit a very simple structure containing highly degenerate ``super-multiplets''. The algebra underlying this structure is identified and several sets of raising and lowering operators are given explicitely. On the basis of this algebra and numerical studies we give the complete spectrum and thermodynamics of the $SU(2)$ system.Comment: 9 pages, late

On the superintegrability of the rational Ruijsenaars-Schneider model

The rational and hyperbolic Ruijsenaars-Schneider models and their non-relativistic limits are maximally superintegrable since they admit action variables with globally well-defined canonical conjugates. In the case of the rational Ruijsenaars-Schneider model we present an alternative proof of the superintegrability by explicitly exhibiting extra conserved quantities relying on a generalization of the construction of Wojciechowski for the rational Calogero model.Comment: added 2 references and some comments in v2, 10 page

On frequencies of small oscillations of some dynamical systems associated with root systems

In the paper by F. Calogero and author [Commun. Math. Phys. 59 (1978) 109-116] the formula for frequencies of small oscillations of the Sutherland system ($A_l$ case) was found. In present note the generalization of this formula for the case of arbitrary root system is given.Comment: arxiv version is already officia

N-Soliton Solutions to a New (2 + 1) Dimensional Integrable Equation

We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation, $\phi_{xt} + \phi_{xxxz}/4 + \phi_x \phi_{xz} + \phi_{xx} \phi_z/2 + \partial_x^{-1} \phi_{zzz}/4 = 0$. This equation is obtained by unifying two directional generalization of the KdV equation, composing the closed ring with the KP equation and Bogoyavlenskii-Schiff equation. We also find the Miura transformation which yields the same ring in the corresponding modified equations.Comment: 7 pages, uses ioplppt.st

Generalization of a result of Matsuo and Cherednik to the Calogero-Sutherland- Moser integrable models with exchange terms

A few years ago, Matsuo and Cherednik proved that from some solutions of the Knizhnik-Zamolodchikov (KZ) equations, which first appeared in conformal field theory, one can obtain wave functions for the Calogero integrable system. In the present communication, it is shown that from some solutions of generalized KZ equations, one can construct wave functions, characterized by any given permutational symmetry, for some Calogero-Sutherland-Moser integrable models with exchange terms. Such models include the spin generalizations of the original Calogero and Sutherland ones, as well as that with $\delta$-function interaction.Comment: Latex, 7 pages, Communication at the 4th Colloquium "Quantum Groups and Integrable Systems", Prague (June 1995