( to appear in Phys. Lett.B )

We study a multispecies one-dimensional Calogero model with two- and three-body interactions. Using an algebraic approach (Fock space analysis), we construct ladder operators and find infinitely many, but not all, exact eigenstates of the model Hamiltonian. Besides the ground state energy, we deduce energies of the excited states. It turns out that the spectrum is linear in quantum numbers and that the higher-energy levels are degenerate. The dynamical symmetry responsible for degeneracy is SU(2). We also find the universal critical point at which the model exhibits singular behaviour. Finally, we make contact with some special cases mentioned in the literature.Comment: 16 pages, no figures, LateX, enlarged version that appeared in Phys.Lett.

A unified view of general multimode oscillator algebras with Fock-like representations is presented.It extends a previous analysis of the single-mode oscillator algebras.The expansion of the a i a † j operators is extended to include all normally ordered terms in creation and annihilation operators and we analyze their action on Focklike states.We restrict ourselves to the algebras compatible with number operators.The connection between these algebras and generalized statistics is analyzed.We demonstrate our approach by considering the algebras obtainable from the generalized Jordan-Wigner transformation, the para-Bose and para-Fermi algebras, the Govorkov "paraquantization" algebra and generalized quon algebra.Recently,much attention has been devoted to the study of quantum groups and algebras 1 , noncommutative spaces and geometries 2 ,generalized notions of symmetries as well as to their diverse applications in physics 3 . These approaches are in close relationship with study of deformed oscillators,algebras and their Fock representations. Single-mode oscillator algebras were studied by a number of authors 4 .A unified view of single-mode oscillator algebras was proposed by Bonatsos and Daskaloyannis 5 and Meljanac et al. 6 . Multimode oscillator algebras are much more complicated and only partial results exist in the literature 7 .Particularly,the R-matrix approach to multimode algebras was followed in Refs.(8).Some of multimode algebras,but not all, can be obtained from an ordinary Bose algebra with equal number of oscillators 9 . On the other hand, there is an old additional physical motivation to study multimode oscillator algebras,connected with the problem of generalized statistics,different from Bose and Fermi statistics 10 .The first consistent example of it was para-Bose and para-Fermi statistics 11,12 and, recently, a new paraquantization 13 satisfying trilinear commutation relations between annihilation and creation operators. These types of statistics are characterized by a discrete parameter called the order of parastatistics p ∈ N,interpolating between Bose and Fermi statistics.Recently,a new interpolation,namely infinite quon statistics characterized by a continuous parameter, has been proposed and analyzed 14,15 .Its multiparameter extension was studied in Refs.(16).Parastatistics can be applied in spaces with an arbitrary number of dimensions and those statistics with a continuous parameter could be relevant to lower dimensions.Specially,in the (2 + 1) dimensional space,another

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 nonhermitean number operators. The results of a general analysis are applied to the S M -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,

We define a new multispecies model of Calogero type in D dimensions with harmonic, two-body and three-body interactions. Using the underlying conformal SU(1,1) algebra, we indicate how to find the complete set of the states in Bargmann-Fock space. There are towers of states, with equidistant energy spectra in each tower. We explicitely construct all polynomial eigenstates, namely the center-of-mass states and global dilatation modes, and find their corresponding eigenenergies. We also construct ladder operators for these global collective states. Analysing corresponding Fock space, we detect the universal critical point at which the model exhibits singular behavior. The above results are universal for all systems with underlying conformal SU(1,1) symmetry.Comment: 14 pages, no figures, to be published in Phys.Lett.