Resonance overlap structure in the microwave ionization of the hydrogen atom

The microwave ionization of the hydrogen atom involves most of the open issues concerning classical and quantum chaos. Much recent research has considered quasi-one-dimensional extended states for which ionization thresholds have been estimated using a classical picture in which ionization proceeds through the overlap of an infinity of nonlinear resonances. Using a canonical transformation to Deprit's ``Lissajous elements'' which makes the two-dimensional nature of the problem explicit, an accurate and improved ionization threshold, compared to previous resonance overlap criteria, is obtained through the overlap of only two nonlinear resonances in the one-dimensional limit.