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arXiv:hep-th/9603050

On the solution of the Calogero model and its generalization to the case of distinguishable particles

Alain Dasnières de Veigy

Published Mar 8, 1996• 3 citations
arXivHTMLPDF

Abstract

The 3-body Calogero problem is solved by separation of variables for arbitrary exchange statistics. A numerical computation of the 4-body spectrum is also presented. The results display new features in comparison with the standard case of bosons and fermions, for instance the energies are not linear with the interaction parameter $ν$ and Bethe ansatz as well as Haldane's statistics are not verified.

DOI
10.48550/arxiv.hep-th/9603050

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The 3-body Calogero problem is solved by separation of variables for arbitrary exchange statistics. A numerical computation of the 4-body spectrum is also presented. The results display new features in comparison with the standard case of bosons and fermions, for instance the energies are not linear with the interaction parameter ννν and Bethe ansatz as well as Haldane's statistics are not verified.
We show that the thermodynamic Bethe ansatz equations for one-dimensional integrable many-body systems can be reinterpreted in such a way that they only code the statistical interactions, in the sense of Haldane, between particles of identical or different momenta. Thus, the thermodynamic properties of these systems can be characterized by the generalized ideal gases recently proposed by one of us. For example, the Yang-Yang δδδ-function gas is a gas with specific statistical interactions between particles of different momenta, while the Calogero-Sutherland system provides a model for an ideal gas of particles with a fractional statistics.
This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.
Following the definition of Haldane's exclusion statistics, we propose a new and simple representation of the spectrum and the state counting of the quantum Calogero model. We show that N non-interacting particles with statistical interactions confined in an external harmonic well give exactly the same energy spectrum and the same degeneracy as those of the N -particle quantum Calogero model. We also show that this picture exactly corresponds to the generalized ideal gas, which was introduced by Bernard and Wu.