  We consider a restricted three body problem, where two interacted particles
are located in two dimensional (2D) plane and interact with the third one
located in the parallel spatially separated plane. The system of such type can
be formed in the semiconductor coupled quantum wells, where the electrons (or
holes) and direct excitons spatially separated in different parallel
neighboring quantum wells that are sufficiently close to interact and form
negative X- or positive X+ indirect trions. It is shown that at large interwell
separations, when the interwell separation much greater than the exciton Bohr
radius, this problem can be solved analytically using the cluster approach.
Analytical results for the energy spectrum and the wave functions of the
spatially indirect trion are obtained, their dependence on the interwell
separations is analyzed and a conditional probability distribution is
calculated. The formation of 2D Wigner crystal of trions at the low densities
is predicted. It is shown that the critical density of the formation of the
trion Wigner crystal is sufficiently greater than the critical density of the
electron Wigner crystal.
