Kyoto University Separation of variables in the $A_{2} $ type Jack polynomials

An integral operator $M $ is constructed performing a separation of variables for the 3-particle quantum Calogero-Sutherland $(\mathrm{C}\mathrm{S}) $ model. Under the action of $M $ the CS eigenfunctions (Jack polynomials for the root system $A_{2} $ ) are transformed to the factorized form $\varphi(y_{1})\varphi(y_{2}) $ , where $\varphi(y) $ is a trigonometric polynomial of one variable expressed in terms of the $\mathrm{s}F_{2} $ hypergeometric series. The inversion of $M $ produces a new integral representation for the $A_{2} $ Jack polynomials. 1 Quantum Calogero-Sutherland model Define $N $ differential operators $\{H_{k}\}_{k1}^{N}= $ ’ acting on functions of $N $ variables $q^{arrow}=$ $\{q_{1}, \ldots, q_{N}\} $ and depending on a parameter $g $ , by the formula [1] $H_{k^{\wedge}}=0 \leq l\leq\sum_{\frac{k}{2}\sigma\in}\sum \mathfrak{S}_{N}\frac{1}{\ # G(l,k-2l)}D_{l,k-}^{\sigma}2l$ (1) where $D_{\gamma \mathrm{t},n},=u(q_{1^{-}}q2)u(q3-q4) \ldots u(q_{2}m-1-q2\gamma n),\frac{(-\dot{i})^{n}\partial^{n}}{\partial q_{2n+1}\partial q_{2}?n+2\cdots\partial q2m+n}$. (2) Here we denote $u(q)=-g(g-1)/\sin^{2}q $ , whereas $\mathfrak{S}_{N} $ is the permutation grou