Multipole radiation in a collisonless gas coupled to electromagnetism or scalar gravitation

We consider the relativistic Vlasov-Maxwell and Vlasov-Nordstr\"om systems which describe large particle ensembles interacting by either electromagnetic fields or a relativistic scalar gravity model. For both systems we derive a radiation formula analogous to the Einstein quadrupole formula in general relativity.Comment: 21 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.-

Goldfish geodesics and Hamiltonian reduction of matrix dynamics

We relate free vector dynamics to the eigenvalue motion of a time-dependent real-symmetric NxN matrix, and give a geodesic interpretation to Ruijsenaars Schneider models.Comment: 8 page

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

Invariants of the Haldane-Shastry SU(N)SU(N) Chain

Using a formalism developed by Polychronakos, we explicitly construct a set of invariants of the motion for the Haldane-Shastry $SU(N)$ chain.Comment: 11 pages, UVA-92-0

Lax pairs, Painlev\'e properties and exact solutions of the alogero Korteweg-de Vries equation and a new (2+1)-dimensional equation

We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV) equation. Moreover, we modify the T operator in the the Lax pair of the CKdV equation, in the search of a (2+1)-dimensional case and thereby propose a new equation in (2+1) dimensions. We named this the (2+1)-dimensional CKdV equation. We show that the CKdV equation as well as the (2+1)-dimensional CKdV equation are integrable in the sense that they possess the Painlev\'e property. Some exact solutions are also constructed

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

Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV

We consider multiple lattices and functions defined on them. We introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow varying lattices. We use these results to perform the multiple--scale reduction of the lattice potential Korteweg--de Vries equation.Comment: 17 pages. 1 figur

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

Finite Chern-Simons matrix model - algebraic approach

We analyze the algebra of observables and the physical Fock space of the finite Chern-Simons matrix model. We observe that the minimal algebra of observables acting on that Fock space is identical to that of the Calogero model. Our main result is the identification of the states in the l-th tower of the Chern-Simons matrix model Fock space and the states of the Calogero model with the interaction parameter nu=l+1. We describe quasiparticle and quasihole states in the both models in terms of Schur functions, and discuss some nontrivial consequences of our algebraic approach.Comment: 12pages, jhep cls, minor correction