Inelastic transitions in collisions of multidimensional harmonic oscillators

A Feynman path integral representation for the $S$ matrix of the problem is used to obtain an exact expression for the transition probabilities when multidimensional harmonic oscillators collide under the influence of a perturbation that contains terms linear and quadratic in the coordinates. The possibility of transition to the approximation of a “generalized external force” is analyzed. As an example, the problem of the interaction of two one-dimensional harmonic oscillators with a potential that depends on the time through the inverse hyperbolic cosine is solved.