  Cross sections for $^{40}$Ca + $\alpha$ at low energies have been calculated
from two different models and three different $\alpha$-nucleus potentials. The
first model determines the cross sections from the barrier transmission in a
real nuclear potential. Second, cross sections are derived within the optical
model using a complex nuclear potential. The excitation functions from barrier
transmission are smooth whereas the excitation functions from the optical model
show a significant sensitivity to the chosen imaginary potential. Cross
sections far below the Coulomb barrier are lower from barrier transmission than
from the optical model. This difference is explained by additional absorption
in the tail of the imaginary part of the potential in the optical model. At
higher energies the calculations from the two models and all $\alpha$-nucleus
potentials converge. Finally, in contradiction to another recent study where a
double-folding potential failed in a WKB calculation, the applicability of
double-folding potentials for $^{40}$Ca + $\alpha$ at low energies is clearly
confirmed in the present analysis for the simple barrier transmission model and
for the full optical model calculation.
