The quantum diffusion approach is extended to low energy fusion (capture) reactions of light- and medium-mass nuclei. The dependence of the friction parameter on bombarding energy is taken into account. A simple analytic expression is obtained for the capture probability at extreme sub-barrier energies. The calculated cross-sections are in a good agreement with the experimental data. The fusion excitation functions calculated within the quantum diffusion and WKB approaches are compared and presented in the astrophysical $S$-factor representation.
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