# Calogero-Moser eigenfunction

The folllowing function $$J(t_1,t_2,t_3,m,h)=[(1-e^{t_1-t_2})(1-e^{t_2-t_3})(1-e^{t_1-t_3})]^{-m/h} e^{-\frac{a_1t_1+a_2t_2+a_3t_3}{h}}\sum_{k_{1,1},k_{2,1},k_{2,2}\ge0}e^{(t_1-t_2)k_{1,1}+(t_2-t_3)(k_{2,1}+k_{2,2})}\prod_{i=1}^2\prod_{j=1}^2\frac{(1-\frac{a_i}{h}+\frac{a_j}{h}+\frac{m}{h})_{k_{2,i}-k_{2,j}}}{(-\frac{a_i}{h}+\frac{a_j}{h})_{k_{2,i}-k_{2,j}}}\prod_{i=1}^2\frac{(-\frac{a_1}{h}+\frac{a_i}{h}-\frac{m}{h})_{k_{1,1}-k_{2,i}}}{(1-\frac{a_1}{h}+\frac{a_i}{h})_{k_{1,1}-k_{2,i}}} \prod_{i=1}^2\prod_{j=1}^3\frac{(-\frac{a_i}{h}+\frac{a_j}{h}-\frac{m}{h})_{k_{2,i}}}{(1-\frac{a_i}{h}+\frac{a_j}{h})_{k_{2,i}}}$$ is supposed to be an eigenvalue of the Calogero-Moser Hamiltonian $$H=\sum_{i=1}^3\frac{\partial^2}{\partial t_i^2}-2\frac{m}{h}(\frac{m}{h}+1)\Bigg[\frac{e^{t_1-t_2}}{(1-e^{t_1-t_2})^2}+\frac{e^{t_2-t_3}}{(1-e^{t_2-t_3})^2}+\frac{e^{t_1-t_3}}{(1-e^{t_1-t_3})^2} \Bigg]$$ with an eigenvalue $$HJ=\Bigg[\frac{a_1^2+a_2^2+a_3^2}{h^2}\Bigg]J$$ Note that we use the Pochhammer symbol $(x)_k$ defined by $$(x)_k = \left\{ \begin{array}{cc} \prod_{i=0}^{k-1} (x+i) & \,\,\text{for}\,\, k>0\\ 1 & \,\,\text{for}\,\, k=0\\ \prod_{i=1}^{k} \dfrac{1}{x-i} & \,\,\text{for}\,\, k<0 \end{array} \right.$$ I would like hopefully to prove, or at least to check that the function $J$ is an eigenfunction of the Calogero-Moser Hamiltonian $H$ for some powers of $e^{t_i}$. However, the expression is too complicated so that I don't even know how I can check it by Mathematica. From this link, you can download a file including the expression of the function $J$. I wonder if somebody could tell me a good way to prove or to check the equation.

PS) I also posted the same question to mathematica stackexchange since I thought an expert in Mathematica may be able to help me.

Here's the function $J$; paste this in to Mathematica:

J[t1_,t2_,t3_,h_,m_,p_,q_,r_]:=((1-E^(t1-t2))(1-E^(t2-t3))(1-E^(t1-t3)))^(-m/h)E^(-(Subscript[a, 1]t1+Subscript[a, 2]t2+Subscript[a, 3]t3)/(h))Sum[E^((t1-t2)(Subscript[k, 1,1])+(t2-t3)(Subscript[k, 2,1]+Subscript[k, 2,2]))(\!$$\*UnderoverscriptBox[\(\[Product]$$, $$j = 1$$, $$2$$]$$\*UnderoverscriptBox[\(\[Product]$$, $$i = 1$$, $$2$$]
\*FractionBox[$$Pochhammer[\(- \*SubscriptBox[\(a$$, $$i$$]\)/h +
\*SubscriptBox[$$a$$, $$j$$]/h + m/h + 1,
\*SubscriptBox[$$k$$, $$2, i$$] -
\*SubscriptBox[$$k$$, $$2, j$$]]\), $$Pochhammer[\(- \*SubscriptBox[\(a$$, $$i$$]\)/h +
\*SubscriptBox[$$a$$, $$j$$]/h,
\*SubscriptBox[$$k$$, $$2, i$$] -
\*SubscriptBox[$$k$$, $$2, j$$]]\)]\)\))(\!$$\*UnderoverscriptBox[\(\[Product]$$, $$j = 1$$, $$2$$]
\*FractionBox[$$Pochhammer[\(- \*SubscriptBox[\(a$$, $$1$$]\)/h +
\*SubscriptBox[$$a$$, $$j$$]/h - m/h,
\*SubscriptBox[$$k$$, $$1, 1$$] -
\*SubscriptBox[$$k$$, $$2, j$$]]\), $$Pochhammer[1 - \*SubscriptBox[\(a$$, $$1$$]/h +
\*SubscriptBox[$$a$$, $$j$$]/h,
\*SubscriptBox[$$k$$, $$1, 1$$] -
\*SubscriptBox[$$k$$, $$2, j$$]]\)]\))(\!$$\*UnderoverscriptBox[\(\[Product]$$, $$ell = 1$$, $$3$$]$$\*UnderoverscriptBox[\(\[Product]$$, $$i = 1$$, $$2$$]
\*FractionBox[$$Pochhammer[\(- \*SubscriptBox[\(a$$, $$i$$]\)/h +
\*SubscriptBox[$$a$$, $$ell$$]/h - m/h,
\*SubscriptBox[$$k$$, $$2, i$$]]\), $$Pochhammer[1 - \*SubscriptBox[\(a$$, $$i$$]/h +
\*SubscriptBox[$$a$$, $$ell$$]/h,
\*SubscriptBox[$$k$$, $$2, i$$]]\)]\)\)),{Subscript[k, 1,1],0,p},{Subscript[k, 2,1],0,q},{Subscript[k, 2,2],0,r}]

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