Stationary Phase Method for Path Integrals

Both difficulties can be overcome by the method originally introduced in the quantum field theory (Popov 1983) and applied later to the quasiclassical theory of molecule-molecule and gas-surface scattering (Dubrovskiy and Bogdanov 1979a Bogdainov 1980 Bogdanov et al. 1989). 

In case of gas-surface scattering, the stationary phase trajectories in th path integrals (1.2.19) can be fixed both in time and in space by using the following unity decompositions  

After that each of the integrals is evaluated within the quasiclassical method separately before and after fixing point t = 0. The influence functional F[F, F ] at each trajectory branch is considered as a pre-exponential factor and is replaced with its value at the classical trajectory, obtained by the variation of S — S. In this case the classical trajectories r t),r (t) aie real and coincide, but they have jumps in z and P at = 0 (while the trajectories of the generalized eikonal method have jumps only in p). 

This method allows one to take into account the influence of crystal on adatom and can be a starting point for perturbation theory, that would advance this approximation. Of course, it is valid only provided F varies slower than exp(iS), but this question is beyond the present discussion. 

If surface corrugation is neglected, the potential in the equations of motion depends only on z, and the tangential motion is free at each brandi of the trajectory. The classical trajectory R t),z(t) is then found from the equations of motion involving the only potential Vq with boundaiy conditions mgi( 0) = 0, H( 0) = JZo fngi - Pf,i at t - oo  

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