Askold M. Perelomov

A perturbative approach to the quantum elliptic Calogero-Sutherland model

Abstract We solve perturbatively the quantum elliptic Calogero–Sutherland model in the regime in which the quotient between the real and imaginary semiperiods of the Weierstrass P function is small.

An elementary construction of lowering and raising operators for the trigonometric Calogero-Sutherland model

The quantum Calogero–Sutherland model of An-type (Calogero F 1971 J. Math. Phys. 12 419–36, Sutherland B 1972 Phys. Rev. A 4 2019–21) is completely integrable (Olshanetsky M A and Perelomov A M 1977 Lett. Math. Phys. 2 7–13, Olshanetsky M A and Perelomov A M 1978 Funct. Anal. Appl. 12 121–8, Olshanetsky M A and Perelomov A M 1983 Phys. Rep. 94 313–404). Using this fact, we give an elementary construction of lowering and raising operators for the trigonometric case. This is similar to, but more complicated (due to the fact that the energy spectrum is not equidistant) than the construction for the rational case (Perelomov A M 1976 ITEP Preprint No 27).

EXPLICIT SOLUTION OF THE QUANTUM THREE-BODY CALOGERO-SUTHERLAND MODEL

The class of quantum integrable systems associated with root systems was introduced by Olshanetsky and Perelomov as a generalization of the Calogero - Sutherland systems. It was recently shown by one of the authors that for such systems with a potential , the series in the product of two wavefunctions is the -deformation of the Clebsch - Gordan series. This yields recursion relations for the wavefunctions of those systems and, related to them, for generalized zonal spherical functions on symmetric spaces. In this letter this approach is used to compute the explicit expressions for the three-body Calogero - Sutherland wavefunctions, which are the Jack polynomials. We conjecture that similar results are also valid for the more general two-parameters deformation (-deformation) introduced by Macdonald.

DERIVATIVES OF GENERALIZED GEGENBAUER POLYNOMIALS

We prove some new formulas for the derivatives of the generalized Gegenbauer polynomials associated with the Lie algebra A2.

On an approach for computing the generating functions of the characters of simple Lie algebras

We describe a general approach to obtain the generating functions of the characters of simple Lie algebras which is based on the theory of the quantum trigonometric Calogero–Sutherland model. We show how the method works in practice by means of a few examples involving some low rank classical algebras.

The Hamiltonian of the quantum trigonometric Calogero–Sutherland model in the exceptional algebra E8

We express the Hamiltonian of the quantum trigonometric Calogero–Sutherland model for the Lie algebra E8 and coupling constant κ by using the fundamental irreducible characters of the algebra as dynamical independent variables.

The generating function for a particular class of characters of SU(n)

We compute the generating function for the characters of the irreducible representations of SU(n) whose associated Young diagrams have only two rows with the same number of boxes. The result is given by formulae (11), (14), (25)–(27) and is a rational determinantal expression in which both the numerator and the denominator have a simple structure when expressed in terms of Schur polynomials.