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SU(1, 1) algebra and generalized Calogero model

The rational Calogero model is one of the most famous and exaustively studied examples of integrable systems. Since its inception more than thirty years ago [1], the model and its various descendants continue to be of interest for both physics and mathematics community. The model describes $N$ identical (single-species) particles on the line which interact through an inverse-square two-body interaction and are subjected to a common confining harmonic force. The inverse-square potential can be regarded as a pure statistical interaction and the model maps to an ideal gas of particles obeying fractional Haldane statistics [2]. However, in Haldane's formulation of statistics there is a possibility of having particles of different species with a mutual statistical coupling parameter depending on the species coupled. This suggests the generalization of the single-species Calogero model to the multispecies Calogero model. Distinguishabillity of the species can be introduced by allowing particles to have different masses and different couplings to each other. While the single-species Calogero model is completely solvable, very little is known about spectra and wave functions of the multispecies Calogero model. Recently, we used an operator method to analyse a one-dimensional multispecies Calogero model with two- and three-body interactions [3]. We succeeded in finding a class of, but not all, exact eigenstates and eigenenergies of the model Hamiltonian. The analysis relied heavily on the SU(1, 1) algebraic structure of the Hamiltonian and once more stressed the importance of the conformal symmetry of the quantum singular oscillator [4]. The universal $SU(1, 1)$ structure of the model permitted us to discuss and partially solve the problem of $F$ interacting families of Calogero type-particles in one dimension [5]. By imposing the conditions for the absence of the three-body interaction, we found certain relations between the coupling constants, sometimes termed as "weak-strong duality".\\ We were also able to generalize the model to arbitrary dimensions [6]. In spite of the complications caused by inevitable appearance of the three-body interaction, we have found ground state and some of the excited states describing the global collective modes.\\ We emphasize that our algebraic analysis is applicable to all systems with underlying $SU(1, 1)$ algebra.

Calogero model; spectrum generating algebra

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