arXiv:solv-int/9508002v1 21 Aug 1995. UTMS 95-10 .... (21). The conjectures 1 and 2 are proved in the next section for the N = 2 case and in the sections 4 and ...
UTMS 95-10 UAMS 95-06 solv-int/9508002
Separation of variables in the A2 type Jack polynomials
arXiv:solv-int/9508002v1 21 Aug 1995
V.B. Kuznetsov Faculteit voor Wiskunde en Informatica, Universiteit van Amsterdam, Plantage Muidegracht 24, 1018 TV Amsterdam, The Netherlands1 E.K. Sklyanin Department of Mathematical Sciences, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan2
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 A2 ) are transformed to the factorized form ϕ(y1 )ϕ(y2), where ϕ(y) is a trigonometric polynomial of one variable expressed in terms of the 3 F2 hypergeometric series. The inversion of M produces a new integral representation for the A2 Jack polynomials.
1
Quantum Calogero-Sutherland model
Define N differential operators {Hk }N q = k=1 , acting on functions of N variables ~ {q1 , . . . , qN } and depending on a parameter g, by the formula [1] Hk =
X
0≤l≤ k2
X
σ∈SN
1 Dσ #G(l, k − 2l) l,k−2l
(1)
where Dm,n = u(q1 − q2 )u(q3 − q4 ) . . . u(q2m−1 − q2m )
(−i)n ∂ n . ∂q2m+1 ∂q2m+2 . . . ∂q2m+n
(2)
Here we denote u(q) = −g(g − 1)/ sin2 q, whereas SN is the permutation group σ of the set {1, . . . , N}, and G(m, n) = {σ ∈ SN |Dm,n = Dm,n }. 1
On leave from Department of Mathematical and Computational Physics, Institute of Physics, St. Petersburg University, St. Petersburg 198904, Russia. 2 On leave from Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia.
1
Note that, when g → 0, the operators Hk behave as X
Hk = (−i)k
j1
