Low-energy asymptotics of the phases of two-dimensional scattering of a quantum particle by a central long-range potential

We study the two-dimensional scattering of a quantum particle in the field of a central long-range potential decaying in the limit of a large distance $r$ as a power-law function $r^{-\beta}$ with $\beta>2$. We find the explicit low-energy asymptotic forms of all partial scattering phases of such a particle.