Integrable Systems for Particles with Internal Degrees of Freedom

We show that a class of models for particles with internal degrees of freedom are integrable. These systems are basically generalizations of the models of Calogero and Sutherland. The proofs of integrability are based on a recently developed exchange operator formalism. We calculate the wave-functions for the Calogero-like models and find the ground-state wave-function for a Calogero-like model in a position dependent magnetic field. This last model might have some relevance for matrix models of open strings.Comment: 10 pages, UVA-92-04, CU-TP-56

A discrete linearizability test based on multiscale analysis

In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A1, A2 and A3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A3 C-integrability conditions can be linearized by a Möbius transformation

Exact Solution of a N-body Problem in One Dimension

Complete energy spectrum is obtained for the quantum mechanical problem of N one dimensional equal mass particles interacting via potential $$V(x_1,x_2,...,x_N) = g\sum^N_{i < j}{1\over (x_i-x_j)^2} - {\alpha\over \sqrt{\sum_{i < j} (x_i-x_j)^2}}$$ Further, it is shown that scattering configuration, characterized by initial momenta $p_i (i=1,2,...,N)$ goes over into a final configuration characterized uniquely by the final momenta $p'_i$ with $p'_i=p_{N+1-i}$.Comment: 8 pages, tex file, no figures, sign in the first term on the right hand side of eq.3 correcte

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.

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

Goldfishing by gauge theory

A new solvable many-body problem of goldfish type is identified and used to revisit the connection among two different approaches to solvable dynamical systems. An isochronous variant of this model is identified and investigated. Alternative versions of these models are presented. The behavior of the alternative isochronous model near its equilibrium configurations is investigated, and a remarkable Diophantine result, as well as related Diophantine conjectures, are thereby obtained.Comment: 22 page

N=4 supersymmetric 3-particles Calogero model

We constructed the most general N=4 superconformal 3-particles systems with translation invariance. In the basis with decoupled center of mass the supercharges and Hamiltonian possess one arbitrary function which defines all potential terms. We have shown that with the proper choice of this function one may describe the standard, $A_2$ Calogero model as well as $BC_2, B_2,C_2$ and $D_2$ Calogero models with N=4 superconformal symmetry. The main property of all these systems is that even with the coupling constant equal to zero they still contain nontrivial interactions in the fermionic sector. In other words, there are infinitely many non equivalent N=4 supersymmetric extensions of the free action depending on one arbitrary function. We also considered quantization and explicitly showed how the supercharges and Hamiltonian are modified.Comment: 13 pages, LaTeX file, PACS: 11.30.Pb, 03.65.-