Two-dimensional motion of a slow quantum particle in the field of a central long-range potential

We study the two-dimensional motion of a slow quantum particle in the field of a central long-range potential decaying in the limit of long distances $r$ as the power $r^{-\beta}$ with the exponent $\beta\in(1,2)$. We find the low-temperature asymptotic behavior for all partial phases and the differential cross section of the particle scattering and derive a rather simple approximation for the weakly bound state energy.