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, A₂ Calogero model as well as G₂ and BC₂ Calogero models, which, by construction, possess 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. In the quantum case the constructed systems exhibit only invariance with respect to N=4 Poincaré supersymmetry.     

The spherical sector of the Calogero model as a reduced matrix model

We investigate the matrix-model origin of the spherical sector of the rational Calogero model and its constants of motion. We develop a diagrammatic technique which allows us to find explicit expressions of the constants of motion and calculate their Poisson brackets. In this way we obtain all functionally independent constants of motion to any given order in the momenta. Our technique is related to the valence-bond basis for singlet states.     

Multicenter Higgs oscillators and Calogero model

We show that the spherical part of N-particle Calogero model describes, after exclusion of the center of mass, the motion of the particle on (N − 2)-dimensional sphere interacting with N(N − 1)/2 force centers with Higgs oscillator potential. In the case of four-particle system these force centers are located at the vertexes of cuboctahedron. The geometry of the five-particle case is also investigated.     

Fun and frustration with Calogero model

We review some algebraical (oscillator) aspects of N-body single-species and multispecies Calogero models in one dimension. We treat them as a particular cases of deformed harmonic oscillators and discuss the corresponding Fock spaces. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Constants of motion of the four-particle Calogero model

We present the explicit expressions of the complete set of constants of motion of four-particle Calogero model with excluded center of mass, i.e. of the A ₃ rational Calogero model. Then we find the constants of motion of its spherical part, defining two-dimensional 12-center spherical oscillator, with the force centers located at the vertexes of cuboctahedron.     

Density waves in the Calogero model - revisited

The Calogero model bears, in the continuum limit, collective excitations in the form of density waves and solitary modulations of the density of particles. This sector of the spectrum of the model was investigated, mostly within the framework of collective-field theory, by several authors, over the past 15 years or so. In this work we shall concentrate on periodic solutions of the collective BPS-equation (also known as “finite amplitude density waves”), as well as on periodic solutions of the full static variational equations which vanish periodically (also known as “large amplitude density waves”). While these solutions are not new, we feel that our analysis and presentation add to the existing literature, as we explain in the text. In addition, we show that these solutions also occur in a certain two-family generalization of the Calogero model, at special points in parameter space. A compendium of useful identities associated with Hilbert transforms, including our own proofs of these identities, appears in Appendix A. In Appendix B we also elucidate in the present paper some fine points having to do with manipulating Hilbert-transforms, which appear ubiquitously in the collective field formalism. Finally, in order to make this paper self-contained, we briefly summarize in Appendix C basic facts about the collective field formulation of the Calogero model.     

Quantization and conformal properties of a generalized Calogero model

We analyze a generalization of the quantum Calogero model with the underlying conformal symmetry, paying special attention to the two-body model deformation. Owing to the underlying SU(1,1) symmetry, we find that the analytic solutions of this model can be described within the scope of the Bargmann representation analysis, and we investigate its dynamical structure by constructing the corresponding Fock space realization. The analysis from the standpoint of supersymmetric quantum mechanics (SUSYQM), when applied to this problem, reveals that the model is also shape invariant. For a certain range of the system parameters, the two-body generalization of the Calogero model is shown to admit a one-parameter family of self-adjoint extensions, leading to inequivalent quantizations of the system. PACS 02.30.Ik; 03.65.Fd; 03.65.-w     

Deformed Heisenberg algebras, a Fock-space representation and the Calogero model

We describe generally deformed Heisenberg algebras in one dimension. The condition for a generalized Leibniz rule is obtained and solved. We analyze conditions under which deformed quantum-mechanical problems have a Fock-space representation. One solution of these conditions leads to a q-deformed oscillator already studied by Lorek et al., and reduces to the harmonic oscillator only in the infinite-momentum frame. The other solution leads to the Calogero model in ordinary quantum mechanics, but reduces to the harmonic oscillator in the absence of deformation. Received: 27 April 2000 / Published online: 8 September 2000     

Permutation invariant algebras, a Fock space realization and the Calogero model

We study permutation invariant oscillator algebras and their Fock space representations using three equivalent techniques, i.e. (i) a normally ordered expansion in creation and annihilation operators, (ii) the action of annihilation operators on monomial states in Fock space and (iii) Gram matrices of inner products in Fock space. We separately discuss permutation invariant algebras which possess hermitean number operators and permutation invariant algebras which possess non-hermitean number operators. The results of a general analysis are applied to the -extended Heisenberg algebra, underlying the M-body Calogero model. Particular attention is devoted to the analysis of Gram matrices for the Calogero model. We discuss their structure, eigenvalues and eigenstates. We obtain a general condition for positivity of eigenvalues, meaning that all norms of states in Fock space are positive if this condition is satisfied. We find a universal critical point at which the reduction of the physical degrees of freedom occurs. We construct dual operators, leading to the ordinary Heisenberg algebra of free Bose oscillators. From the Fock-space point of view, we briefly discuss the existence of a mapping from the Calogero oscillators to the free Bose oscillators and vice versa. Received: 26 July 2001 / Revised version: 9 January 2002 / Published online: 12 April 2002     

Collective field formulation of the multi-species Calogero model and its duality symmetries

We study the collective field formulation of a restricted form of the multi-species Calogero model, in which the three-body interactions are set to zero. We show that the resulting collective field theory is invariant under certain duality transformations, which interchange, among other things, particles and antiparticles, and thus generalize the well known strong-weak coupling duality symmetry of the ordinary Calogero model. We identify all these dualities, which form an Abelian group, and study their consequences. We also study the ground state and small fluctuations around it in detail, starting with the two-species model, and then generalizing to an arbitrary number of species.     

Quantum hydrodynamics, the quantum benjamin-ono equation, and the Calogero model

Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes--density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field--the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.     

Dynamical R-matrices for Calogero models

We present a systematic study of the integrability of the Calogero models, degenerate as well as elliptic, associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs of Lie algebras, where “integrability” is understood to encompass not only the existence of a Lax representation for the equations of motion but also the—more far-reaching—existence of a (dynamical) R-matrix. Using the standard group-theoretical machinery available in this context, we show that integrability of these models, in this sense, can be reduced to the existence of a certain function, denoted here by F, defined on the relevant root system and taking values in the respective Cartan subalgebra, subject to a rather simple set of algebraic constraints: these ensure, in one stroke, the existence of a Lax representation and of a dynamical R-matrix, all given by explicit formulas. We also show that among the simple Lie algebras, only those belonging to the A-series admit a solution of these constraints, whereas the AIII-series of symmetric pairs of Lie algebras, corresponding to the complex Grassmannians SU(p,q)/S(U(p)×U(q)), allows non-trivial solutions when |p−q|⩽1. Apart from reproducing all presently known dynamical R-matrices for Calogero models, our method provides new ones, namely for the degenerate models when |p−q|=1 and for the elliptic models when |p−q|=1 or p=q.     

Generalized Calogero and Toda Models

JETP Letters - A Calogero–Sutherland system with two types of interacting spin variables has been described using the Hitchin approach and quasicompact structure. Complete integrability has...     

Aspects of Generalized Calogero Model

A multispecies model of Calogero type in D 1 dimensions is constructed. The model includes harmonic, two-body and three-body interactions. Using the underlying conformal SU(1,1) algebra, we find the exact eigenenergies corresponding to a class of the exact global collective states. Analyzing corresponding Fock space, we detect the universal critical point at which the model exhibits singular behavior.     

Dunkl operator,integrability, and pairwise scattering in rational Calogero model

The integrability of the Calogero model can be expressed as zero curvature condition using Dunkl operators. The corresponding flat connections are non-local gauge transformations, which map the Calogero wave functions to symmetrized wave functions of the set of N free particles, i.e. it relates the corresponding scattering matrices to each other. The integrability of the Calogero model implies that any k-particle scattering is reduced to successive pairwise scatterings. The consistency condition of this requirement is expressed by the analog of the Yang–Baxter relation.     

Calogero Conjecture and Hubble Law

Acceptance of the Calogero hypothesis on the cosmic origin of quantizationin the framework of Nelson stochastic mechanics would imply that the age ofuniverse is larger than the standard quantum mechanical interpretation of theredshift measurement implies. This is due to variation of h with the radius of theuniverse and thus with time.