On the solution to the separated equation in the 3-particle Calogero-Moser problem

We propose the exact solution of the equation in separated variable which appears in the process of constructing solutions to the quantum Calogero-Moser three-particle problem with elliptic two-particle potential $g(g-1)\wp(q)$. This solution is found for special values of coupling constants $g\in {\mathbb Z}, \, g>1$. It can be used for solving three-paricle CM problem under appropriate boundary conditions.


Introduction
The problem of finding solutions to quantum integrable finite-dimensional systems in many cases still remains unsolved. The empirical constructions of such solutions were important at early stages of the development of the theory of these systems and lead to many important results being applied to trigonometric Calogero-Sutherland-

Moser systems with the Hamiltonian of the form
for N particles in one dimension with the two-body potential given by The coupling constant g is supposed to be real and chosen as g > 1.
It turned out [1] that the ground-state wave function of the trigonometric model is of factorized form and the wave functions of all the excitations can be written as products of this function and multivariable Jack polynomials [4]. These results were also extended to more complicated cases with interaction terms modified by the introduction of more general root systems [4].
The quantum elliptic many-particle problem which also has been proven to be integrable [2,3] is till now quite far from being solved completely. It has two-particle interaction potential of the form where ℘(q) is the Weierstrass elliptic function with two periods ω 1,2 which do not lie on the line in complex plane. The hermiticity of the Hamiltonian implies ω 1 ∈ R, iω 2 ∈ R. The trigonometric case (2) corresponds to infinite complex period.
In the simplest case of N = 2, the eigenvalue problem for the Hamiltonian This fact inspired the authors in the paper [5] to consider the case of general N > 2 and g ∈ Z. It has been proved that the double quasiperiodic solutions for many-particle wave functions are still expressed in terms of the Weierstrass σ functions but the procedure of finding them is rather complicated. They were able to find it explicitly only for N = 3, g = 2. In [7], these solutions have been presented analytically for arbitrary N > 2 and g ≥ 2, also in overcomplicated form requiring many nontrivial operations to their explicit writing. As for arbitrary real g > 1 the solution of the eigenproblem for the elliptic case was constructed by the perturbation theory in the form of infinite series [9].
However, there is another approach to finding the solutions for the dynamics of integrable systems, namely separation of variables. It is well known in its simple form using purely co-ordinate transformations. As for elliptic CM systems. simple forms do not work but separation still take place as it was proposed in [6] for 3particle case at arbitrary values of coupling constant g. The separation of variables occurs after transformation corresponding to a classical canonical transformation of phase space variables mixing coordinates and momenta. The transformation is realized as an integral transform of the wave function in the quantum case. The original two-dimensional problem has been reduced to one-dimensional one and the process of finding the eigenfunction contains investigation of the solution to a third order ordinary differential equation [6], where h 1 , h 2 , h 3 are constants (the values of the integrals of motion).
The aim of this paper is to find the explicit solutions to equation (4). We shall show below that for integer values of g, g ≥ 2 they may be obtained via the solution to the system of g usual transcendental equations.

Finding the solution
It should be noted at first that the coefficients in (4)  The situation is changed drastically if g ∈ Z, g > 1. In this case, the leading singularity of ψ(x) is a pole of the order g − 1 and there are no branch points.
Combining this property with double quasiperiodicity allows one to write down the Hermite-like ansatz for the possible solution to (4) where A is inessential normalization constant, γ and {λ s } are parameters which have to be determined, and σ(x) is the Weierstrass sigma function. It is connected with ℘(x) by the relations where ζ(x) is the Weierstrass ζ function with the property We assume that all λ s are mutually different for s = 1, . . . , g − 1 and different from 0 in T.
By consecutive differentiations of (5), one finds Note that all right-hand sides of these equalities are elliptic functions of the argument x. Substitution of these expressions into (4) yields The function B(x) is elliptic and might have poles up to third order at the points x = 0, x = −λ s (s = 1, ...g − 1). However, the direct inspection of the Laurent decompositions near these points shows that all the coefficients at the terms x −3 , x −2 , (x+λ s ) −3 , (x+λ s ) −2 vanish identically for arbitrary γ and {λ s }. Hence this function can be written in the form where the constant coefficients b 0 , {b s } should obey the relation (statement (III) of par. 20.12 in [8], e. g.). The Laurent decomposition of (7) near the points x = −λ s with the use of (6) allows one to find the coefficients b s explicitly. Due to (7), all of them should vanish. This results in the system of g − 1 transcendental equations to the parameters γ, {λ s }: It remains only to calculate the constant term in (8). Equivalently, we calculate using the Laurent decomposition of (7) near the point x = 0. After long but straightforward calculations (performed by the MATHEMATICA program), one finds the condition ℘(λ s ) + ih 3 = 0.
The algebraic system (10-11) allows one to determine the parameters γ, {λ s } under which the elliptic function B has no poles and equals zero at one point. Then B(x) = 0 due to the Liouville theorem (statement (IV) of par. 20.12 in [8]). The last equation is cubic in γ. This corresponds to three linearly independent solutions to the original equation (4).
Let us summarize our results. We obtained the explicit solutions of the separated equation (4) at integer couplings g which, in its turn, gives the solution to the threeparticle quantum Calogero-Moser problem via the procedure described in [6]. We conjecture that g equations (10-11) determine the g parameters γ, λ s (s = 1, . . . , g − 1) in the generic case at the least. However, it is not clear whether the solution to the above problem in the forms known before [5,7] can be transformed into the forms with separated variables. As [5][6][7], we consider in general singular solutions to the differential equation (4) leaving aside the right physical boundary conditions which are even not known here [6].