Geodesic
Inelastic transitions in collisions of multidimensional harmonic oscillators
Teoretičeskaâ i matematičeskaâ fizika, Tome 68 (1986) no. 2, pp. 255-264

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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. 
@article{TMF_1986_68_2_a8,
     author = {A. I. Denisenko and G. V. Dubrovskiy},
     title = {Inelastic transitions in collisions of multidimensional harmonic oscillators},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {255--264},
     publisher = {mathdoc},
     volume = {68},
     number = {2},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1986_68_2_a8/}
}
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A. I. Denisenko; G. V. Dubrovskiy. Inelastic transitions in collisions of multidimensional harmonic oscillators. Teoretičeskaâ i matematičeskaâ fizika, Tome 68 (1986) no. 2, pp. 255-264. http://geodesic.mathdoc.fr/item/TMF_1986_68_2_a8/