The true ion-ion interaction governed by the repulsive electrostatic potential and the attractive nuclear potential at small separation is approximated as a Coulomb barrier decorated by an attractive $\delta$-shell potential placed under the barrier. This composite potential can generate unbound or resonance states with positive energy eigenvalues. The resonances are determined through the calculation of scattering phase shift $\delta$ of the S matrix. We make the $\delta$-shell potential complex with an attractive imaginary part in it and obtain the phase shift $\delta$ as a complex quantity. This complex phase shift gives nonzero positive reaction cross section to be considered as fusion cross section ${\sigma }_f$ in the calculation. Resonances generated by our model potential combination are clearly manifested as positive peaks in the variation of ${\sigma }_f$ as a function of energy E. The presentation of ${\sigma }_f$ as $\frac{d^2\left(E{\sigma }_f\right)}{d{E}^2}$ in variance with E shows oscillations or structures with several peaks and valleys including valleys of negative values. The calculated results of ${\sigma }_f$ and $\frac{d^2\left(E{\sigma }_f\right)}{d{E}^2}$ are shown to account for the respective measured results showing oscillatory structures with remarkable success in the fusion data of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$, ${}^{12}\mathrm{C}+{}^{16}\mathrm{O}$, ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}+{}^{90,94,96}\mathrm{Zr}$ systems involving light as well as heavy nuclei.

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• Direct reactions
• Nuclear fusion
• Nuclear reactions