## The Calogero-Sutherland Model And Generalized Classical Polynomials (1997)

Venue: | Comm. Math. Phys |

Citations: | 70 - 9 self |

### BibTeX

@ARTICLE{Baker97thecalogero-sutherland,

author = {T. H. Baker and P.J. Forrester},

title = {The Calogero-Sutherland Model And Generalized Classical Polynomials},

journal = {Comm. Math. Phys},

year = {1997},

volume = {188},

pages = {175--216}

}

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### Abstract

this paper. The first is the discussion of some mathematical properties relating to the eigenfunctions, while the second is the evaluation of the density in the ground state and the exact solution of (1.6) for certain initial conditions. These problems are in fact inter-related; we find that the density for each system can be written in terms of a certain eigenstate and that a summation theorem for the eigenstates gives an exact solution of (1.6). A feature of the Schrodinger operators (1.2) is that after conjugation with the ground state: \Gamma e i @ \Gamma fi @W (1.7) the resulting differential operator has a complete set of polynomial eigenfunctions. In Section 2 we consider the form of the expansion of these polynomials in terms of some different bases of symmetric functions. We note that in the N = 1 case, after a suitable change of variables, the operator (1.7) with W given by (1.3) is the eigenoperator for the classical Hermite, Laguerre and Jacobi polynomials. Previous studies of the operator for general N in the Jacobi case [1] have established an orthogonality relation. Since the polynomials in the Hermite and Laguerre cases are limiting cases of these generalized Jacobi polynomials, we can obtain the corresponding orthogonality relations via the limiting procedure. The generalized Hermite polynomials, which are the polynomial eigenfunctions of (1.4) with W = W as given by (1.3a), are studied in Section 3. Many higher-dimensional analogues of properties of the classical Hermite polynomials are obtained, including a generating function formula, differentiation and integration formulas, a summation theorem and recurrence relations. An analogous study of the generalized Laguerre polynomials is performed in Section 4. In Section 5 we relate the...