Amplitude of scattering at high energies on a singular potential of power type

The well-known approximate expressions for the phase shifts $\delta(\lambda,k)$ for the potential $V(r)=g^2r^{-n}$ $(g>0, n>2)$ at high energies are improved by adding to the approximation of small $\lambda$ a certain polynomial in $\lambda$ with subsequent “joining” to the approximation of large $\lambda$. Approximate expressions are obtained for the scattering amplitude $f(\theta, k)$ and the differential cross section $d\sigma/d\theta$ as $k\to\infty$ for different values of $theta$. It is shown that the power potentials are weakly singular in the sense that their singular core, which determines the partial waves with small $\lambda$ and the scattering through large angles, does not influence the total cross section as $k\to\infty$.