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We study the near-threshold behaviour of scattering phase shifts and quantum reflection amplitudes for systems described by a circularly symmetric potential in two spatial dimensions. For $s$ waves, $m=0$, the centrifugal potential is nonvanishing, and the leading near-threshold energy dependence of phase shifts and amplitudes for scattering and quantum reflection is logarithmic. Nevertheless, a scattering length (for potentials falling off faster than $1/r^2$) and an effective range (for potentials falling off faster than $1/r^4$) can be defined. Analytical results are derived for homogeneous potentials, and the necessary modification of the effective-range expansion is given for potentials proportional to $1/r^4$. For $m\ne 0$ we give analytical expressions for the near-threshold behaviour of phase shifts as well as scattering and quantum reflection amplitudes, which are generally valid, even when $m$ is neither integer nor half-integer. |