Analysis of the spatial quantized three-body problem

In this paper, a new version of the restricted three-body problem is considered. That is the spatial quantized restricted three-body problem. The equations of motion of the infinitesimal body are constructed. The equilibria points in plane of primaries motion and out of plane are explored under the quantized gravitational potential effect. The permissible and forbidden regions of motion are also analyzed in both planes, with respect to relative small, middle and large values of the Jacobian constant. We observe that the size of permissible (forbidden) regions of motion are decreasing (increasing) with increasing the Jacobian constant values and vice versa. We demonstrate that the permissible regions of motion shrink with increasing the Jacobian constant value, and the infinitesimal body will be compelled to move in a surrounding region of one the primaries. It will has no free to switch or exchange its motion from one to other. Furthermore, we emphasize that, in spit of quantum corrections are tiny small effect, there is no warranty that the dynamical system properties are unaffected. Because there are changes in the locations of equilibria points and the regions of permissible motion, even which are very small. We think that these corrections may have considerable impact and can be tested within frame of small distances. Substantially, we are determine the parametric evolution of the equilibrium points of the system, along with the energetically allowed and forbidden regions of motion.