Genetic Algorithm based Inverse Potentials for Resonant States of $\alpha-^{12}C$ Using Variable Phase Approach

Elastic scattering between $\alpha$-particles and $^{12}\mathrm{C}$ nuclei plays a crucial role in understanding resonance phenomena in light nuclear systems. In this work, we construct inverse potentials for resonant states in $\alpha$-$^{12}\mathrm{C}$ elastic scattering using the variable phase approach, in tandem with a genetic algorithm based optimization technique. The reference function for the potential in the phase equation is chosen as a combination of three smoothly joined Morse-type functions. The parameters of the reference function are genetically evolved to minimize the the mean squared error (MSE) between the numerically obtained scattering phase shifts and the expected values. The resulting inverse potentials accurately reproduce the resonance energies ($E_r$) and the resonance widths ($\Gamma_r$) for the $\ell^{\pi}$ states, $1^-$, $2^+$, $3^-$, and $4^+$, showing excellent agreement with experimental data. This computational approach to constructing inverse potentials serves as a complementary to conventional direct methods for investigating nuclear scattering phenomena.