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arXiv:1805.09899 · Nuclear Physics B

Mapping the Calogero model on the anyon model

Stéphane Ouvry, Alexios P. Polychronakos

Published Sep 27, 2018• 15 citations
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Abstract

We explicitly map the N-body one dimensional Calogero eigenstates in a harmonic well to the lowest Landau level sector of N-body eigenstates of the two dimensional anyon model in a harmonic well. The mapping is achieved in terms of a convolution kernel that uses as input the scattering eigenstates of the free Calogero model on the infinite line, which are obtained in an operator formulation.

DOI
10.1016/j.nuclphysb.2018.09.011

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We explicitly map the N-body one dimensional Calogero eigenstates in a harmonic well to the lowest Landau level sector of N-body eigenstates of the two dimensional anyon model in a harmonic well. The mapping is achieved in terms of a convolution kernel that uses as input the scattering eigenstates of the free Calogero model on the infinite line, which are obtained in an operator formulation.
A one-dimensional model with F interacting families of Calogero-type particles is studied. The model includes harmonic, two-body and three-body interactions. We emphasize the universal SU(1, 1) structure of the model. We show how SU(1, 1) generators for the whole system are composed of SU(1, 1) generators of arbitrary subsystems. We find the exact eigenenergies corresponding to a class of the exact eigenstates of the F-family model. By imposing the conditions for the absence of the three-body interaction, we find certain relations between the coupling constants.
A similarity transformation is constructed, through which a system of particles interacting with inverse-square two-body and harmonic potentials in one dimension can be mapped identically to a set of free harmonic oscillators. This equivalence provides a straightforward method to find the complete set of eigenfunctions, the exact constants of motion, and a linear ${W}_{1+\ensuremath{\infty}}$ algebra associated with this model. It is also demonstrated that a large class of models with long-range interactions, both in one and higher dimensions can be made equivalent to decoupled oscillators.
We study a one-dimensional model with F interacting families of Calogero-type particles. The model includes harmonic, two-body and three-body interactions. We emphasize the universal SU (1,1) structure of the model. We show how SU (1,1) generators for the whole system are composed of SU (1,1) generators of arbitrary subsystems. We find the exact eigenenergies corresponding to a class of the exact eigenstates of the F-family model. By imposing the conditions for the absence of the three-body interaction, we find certain relations between the coupling constants. Finally, we establish some relations of equivalence between two systems containing F families of Calogero-type particles.