Supplementary quantity

A being a contour of integration encircling t and t but no other transition point, with the integration performed in the direction that makes Kq positive when t and t are real, i.e., when the barrier is superdense, but negative when t and t are complex conjugate, i.e., when the barrier is underdense. 

We emphasize that for the validity of (4.46) with the expressions (4.47a-c) for (2ra+1) the essential restriction is that d2Q2 z)/dz2 must not be too small at the top of the barrier, which means that close to its top the barrier is approximately parabolic, i.e., that the distance from the barrier to the transition points that are not associated with the barrier must be much larger than t — t. However, when the energy is close to the top of the barrier, it is the slight deviation from parabolic shape close to the top that determines the values of the quantities K2n, n 0, and one needs accurate values of these quantities for obtaining accurate values of f in higher orders of the phase-integral approximation. 

The derivative of (f with respect to the energy E will be needed in Chapter 5, and therefore we shall now give formulas for this derivative. From (4.46) and (4.47a-c) one obtains 

According to sections 6.1.27 and 6.3.3 in Abramowitz and Stegun (1965) the argument of the gamma function occurring in (4.49) can be obtained from the formula 

The barrier connection formula presented in this section is valid uniformly for all energies, below and somewhat above the top of the barrier. We would also like to emphasize that while the connection formulas pertaining to a turning point are one-directional (N. Froman 1966a, Froman and Froman 2002), the barrier connection formula (4.30a,b) along with (4.31)-(4.35) is bi-directional. However, when the energy is close to a resonance energy, a careful discussion is required. 

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Kovac, V., Alonso, E., Revilla, E. (1995). The effect of adding supplementary quantities of seeds during fermentation on the phenolic composition of wines. Am. J. Enol. Vitic., 46, 363-367. 

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