Bounds for transition probabilities in collision theory

Upper and lower bounds for transition probabilities in multichannel collision theory are derived, with numerical applications. The approach provides a general scheme for bounding a unitary matrix which depends on a parameter. Particular choices of the parameter include the radial approach coordinate of the colliding systems, with application to close-coupling expansions (both quantum mechanical and semiclassical); the strength parameter of the potential, with applications to the exponential approximation; the time, with applications to transitions induced by a time-dependent potential (a case previously treated by Spruch); and, in general, any parameter in the interaction potential. The method is based on two key steps: (i) the derivation of an integral equation for the dependence of the scattering matrix on the parameter, which can then be bounded, and (ii) the use of the matrix H (all whose elements are unity) as the initial upper bound for the (absolute value of the) scattering matrix. This initial bound is then improved by iterating the integral equation. Computational examples for rotational excitation in an atom-rotor collision are provided.