  The harmonic inversion method is applied in the case of the hydrogen atom in
a magnetic field to extract classical information from the quantum
photo-ionization cross-section. The study is made close to a saddle-node
bifurcation for which the usual semi-classical formulas give diverging
contributions. All quantities (actions, stabilities and Maslov indices) for
real orbits above the bifurcation and for complex ghost orbits below the
bifurcation, are found to be in excellent agreement with the modified
semi-classical predictions based on a normal form approach.
