Approximate solutions to the quantum problem of two opposite charges in a constant magnetic field

• We have studied the quantum model of two charged particles on a plane with opposite charges and a perpendicular constant magnetic field.
• This model is integrable, although not superintegrable.
• The model under study is described by two fourth degree Hill equations, one trigonometric and the other hyperbolic.
• We have considered two distinct approximations that have exact solution.
• We have applied a perturbative method to improve the approximation.

We consider two particles of equal mass and opposite charge in a plane subject to a perpendicular constant magnetic field. This system is integrable but not superintegrable. From the quantum point of view, the solution is given by two fourth degree Hill differential equations which involve the energy as well as a second constant of motion. There are two solvable approximations in relation to the value of a parameter. Starting from each of these approximations, a consistent perturbation theory can be applied to get approximate values of the energy levels and of the second constant of motion.