Two charges on a plane in a magnetic field: II. Moving neutral quantum system across a magnetic field

The moving neutral system of two Coulomb charges on a plane subject to a constant magnetic field B perpendicular to the plane is considered. It is shown that the composite system of finite total mass is bound for any center-of-mass momentum P and magnetic field strength; the energy of the ground state is calculated accurately using a variational approach. Its accuracy is cross-checked in a Lagrange-mesh method for B=1  a.u. and in a perturbation theory at small B and P. The constructed trial function has the property of being a uniform approximation of the exact eigenfunction. For a Hydrogen atom and a Positronium a double perturbation theory in B and P is developed and the first corrections are found algebraically. A phenomenon of a sharp change of energy behavior for a certain center-of-mass momentum and a fixed magnetic field is indicated.