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On the quantum Kepler map

The motion of a 1D hydrogen atom in a high-frequency periodic electric field may be approximated by the Kepler map. The authors compare the solutions of this 'quantum Kepler map' with the numerical solutions of the time-dependent Schrodinger equation using methods making less drastic approximations. They conclude that the quantum Kepler map is a reasonable approximation provided that the field is not strong enough to excite states far from the initial state. They find, however, that it treats the very highly excited states incorrectly and that this leads both to the overestimation of the ionization probability and to nonphysical, wild fluctuations with very small variations in the field frequency.