The DKP equation in presence of a Woods–Saxon potential: Transmission resonances and bound states

In this paper, we solve the Duffin–Kemmer–Petiau (DKP) equation in the presence of the Woods–Saxon potential barrier and well for spin-one particles. We derive the scattering solution in terms of the Gaussian hypergeometric function ₂F₁(a, b; c; z), the regularized Gaussian hypergeometric function ₂F̃₁(a,b;c;z), and the Gamma function Γ(x). Our analysis reveals the presence of transmission resonances. To observe these resonances, we calculate and plot the transmission T and reflection R coefficients for various parameters of the Woods–Saxon potential barrier. Our results are compared with those obtained for the square potential barrier and the cusp potential barrier, which represent limiting cases of the Woods–Saxon potential barrier. Furthermore, we investigate the bound state solutions, determining the critical turning point V_cr to study particle–antiparticle creation and compute the norm N. Finally, we also compare our results with those obtained for the square potential well and the cusp potential well, confirming that pair creation occurs in both potential wells.