Explicit solution of the quantum three-body Calogero-Sutherland model

TL;DR

This paper derives explicit wave function expressions for the quantum three-body Calogero-Sutherland model, identifying them as Jack polynomials, and suggests potential extensions to more general deformations.

Contribution

It provides the first explicit solutions for three-body wave functions in the Calogero-Sutherland model using a recursive approach, linking them to Jack polynomials.

Findings

Explicit three-body wave functions are expressed as Jack polynomials.

Recursion relations for wave functions are established.

Potential extension to Macdonald deformation is conjectured.

Abstract

Quantum integrable systems generalizing Calogero-Sutherland systems were introduced by Olshanetsky and Perelomov (1977). Recently, it was proved that for systems with trigonometric potential, the series in the product of two wave functions is a deformation of the Clebsch-Gordan series. This yields recursion relations for the wave functions of those systems. In this note, this approach is used to compute the explicit expressions for the three-body Calogero-Sutherland wave functions, which are the Jack polynomials. We conjecture that similar results are also valid for the more general two-parameters deformation introduced by Macdonald.