Laplace transforms of the Hulthén Green's function and their application to potential scattering

We derive closed-form representations for the single and double Laplace transforms of the Hulthén Green's function of the outgoing wave multiplied by the Yamaguchi potential and write them in the maximally reduced form. We use the expression for the double transform to compute the low-energy phase shifts for the elastic scattering in the systems $\alpha$–nucleon, $\alpha$–$\text{He}^3$, and $\alpha$–$\text{H}^3$. The calculation results agree well with the experimental data.