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It has been observed in the past that the usual Airy uniform approximation gives probabilities greater than one, especially for near elastic collisions. By mapping the phase onto −ζ cos y + ky + A rather than (1∕3)y^3 − ζy + A one obtains a uniform approximation involving Bessel functions of the first kind, which approaches unity for the elastic collision. This Bessel uniform approximation is no more complicated than the Airy and also gives good agreement with exact quantum results, even if probabilities are large.

the journal of chemical physics

Semiclassical transition probabilities by an asymptotic evaluation of the <i>S</i> matrix for elastic and inelastic collisions. Bessel uniform approximation

Semiclassical transition probabilities by an asymptotic evaluation of the S matrix for elastic and inelastic collisions. Bessel uniform approximation