Separation of variables in the $A_2$ type Jack polynomials

TL;DR

This paper constructs an integral operator that separates variables in the $A_2$ Jack polynomials, transforming complex eigenfunctions into simpler, factorized forms and providing new integral representations.

Contribution

It introduces a novel integral operator for variable separation in $A_2$ Jack polynomials, enabling factorization and new integral representations.

Findings

Operator $M$ achieves separation of variables for $A_2$ Jack polynomials.

Eigenfunctions are transformed into products of one-variable functions.

New integral representations for $A_2$ Jack polynomials are derived.

Abstract

An integral operator $M$ is constructed performing a separation of variables for the 3-particle quantum Calogero-Sutherland (CS) model. Under the action of $M$ the CS eigenfunctions (Jack polynomials for the root system $A_2$) are transformed to the factorized form $\phi(y_1)\phi(y_2)$, where $\phi(y)$ is a trigonometric polynomial of one variable expressed in terms of the ${}_3F_2$ hypergeometric series. The inversion of $M$ produces a new integral representation for the $A_2$ Jack polynomials.