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...

Abstract not found

Abstract not found

Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schrödinger operators for Calogero-Sutherland-type quantum systems. For the generalized Hermite and Laguerre polynomials the multidimensional analogues of many classical results regarding generating functions, differentiation and integration formulas, recurrence relations and summation theorems are obtained. We use this and related theory to evaluate the global limit of the ground state density, obtaining in the Hermite case the Wigner semi-circle law, and to give an explicit solution for an initial value problem in the Hermite and Laguerre case. 1

We investigate some properties of non-symmetric Jack, Hermite and Laguerre polynomials which occur as the polynomial part of the eigenfunctions for certain Calogero-Sutherland models with exchange terms. For the non-symmetric Jack polynomials, the constant term normalization Nη is evaluated using recurrence relations, and Nη is related to the norm for the non-symmetric analogue of the power-sum inner product. Our results for the non-symmetric Hermite and Laguerre polynomials allow the explicit determination of the integral kernels which occur in Dunkl’s theory of integral transforms based on reflection groups of type A and B, and enable many analogues of properties of the classical Fourier, Laplace and Hankel transforms to be derived. The kernels are given as generalized hypergeometric functions based on non-symmetric Jack polynomials. Central to our calculations is the construction of operators ̂ Φ and ̂ Ψ, which act as lowering-type operators for the non-symmetric Jack polynomials of argument x and x 2 respectively, and are the counterpart to the raising-type operator Φ introduced recently by Knop and Sahi. 1