A simple schematic model for a cross section deficit in $^{12}$C+$^{12}$C fusion reactions

A cross section deficit phenomenon has been observed in $^{12}$C+$^{12}$C fusion reactions at astrophysical energies, at which fusion cross sections are suppressed in the off-resonance regions as compared to fusion cross sections for the $^{12}$C+$^{13}$C system. I here construct a simple schematic model which simulates this phenomenon. The model consists of a random matrix Hamiltonian based on the Gaussian Orthogonal Ensemble (GOE), which is coupled to an entrance channel Hamiltonian in the discrete basis representation. I show that the transmission coefficients are almost unity when both the level density and the decay widths of the GOE configurations are large, realizing the strong absorption regime. On the other hand, when these parameters are small, the transmission coefficients are significantly structured as a function of energy. In that situation, the transmission coefficients at resonance energies reach unity in this model, that is consistent with the experimental finding of the cross section deficit.