Abstract
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 . This solution is found for special values of coupling constants . It can be used for solving three-particle Calogero-Moser problem under the appropriate boundary conditions.
Highlights
The problem of finding solutions to quantum integrable finite-dimensional systems in many cases still remains unsolved
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)℘(q)
This solution is found for special values of coupling constants g ∈, g > 1. It can be used for solving three-particle Calogero-Moser problem under the appropriate boundary conditions
Summary
The problem of finding solutions to quantum integrable finite-dimensional systems in many cases still remains unsolved. The reason for the existence of such a solution is based on its “good” analytic properties in a complex plane of the variable q: at integer g there is no branch points and the only singularity is a pole at q = 0 up to the quasiperiodicity This fact inspired the authors in the paper [7] to consider the case of general N > 2 and g ∈. 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] There is another approach to finding the solutions for the dynamics of integrable systems, namely separation of variables. 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
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