Hydrogen atom in a uniform electromagnetic field as an anharmonic oscillator

A nonbijective canonical transformation, the Kustaanheimo-Stiefel transformation, is used to transform the problem of an anharmonic oscillator, connecting it to the problem of the hydrogen atom in a homogeneous and constant electromagnetic field. The Hamiltonian of a hydrogenlike ion in the presence of an electrostatic field in the positive 3-direction is given, and a quantity is introduced in order to treat the electric and magnetic cases in an unified way. The Kustaanheimo-Stiefel transformation is defined and substituted into the previous quantity. The result is applied to the free-atom case, the electric field case, and the magnetic field case.