Transformation of the contour by weak collisions

In the case of weak collisions the change in J is so slight that one may proceed from an integral description of the process to a differential one, just as in Eq. (1.23). However, the kernel of the integral equation (3.26) specified in Eq. (3.28) is different from that in the Feller equation. Thus, the standard procedure described in [20] is more complicated and gives different results (see Appendix 3). The final form of the equation obtained in the limit y — 1, to — 0 with 

This equation may be solved by separation of the variables, which results in formula (A4.12) of Appendix 4 at t = 0. By Fourier transformation of the latter we find the spectrum 

Although from a mathematical point of view formulae (3.34) and (3.40) have little in common, the spectral transformation described by them proceeds in a similar way (Fig. 3.2). Just as with strong collisions, the contour is gradually symmetrized and its centre is shifted to the average frequency coq with an increase in the density. When the spectrum is narrowed (at T 1), its central part ( Aco] 1/tj) takes the following form  

The width of this Lorentzian line is half as large as that found in (3.37). This, however, is not a surprise because the perturbation theory equation (3.23) predicted exactly this difference in the width of the line narrowed by strong and weak collisions. This is the maximal difference expected within the framework of impact theory when the Keilson-Storer kernel is used and 0 y 1. 

It should be noted that the same method for calculation of isotropic scattering spectra is applied to spherical molecules as well. The only difference between linear and spherical molecules is the shape of the static spectrum, while its collapse proceeds in a qualitatively similar way. 

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