Near-threshold scattering, quantum reflection, and quantization in two dimensions

Abstract

We study the near-threshold behavior of scattering phase shifts, quantum reflection amplitudes, and quantization functions for systems described by a circularly symmetric potential in two spatial dimensions. In contrast to the three-dimensional case, the centrifugal potential $({m}^{2}\ensuremath{-}\frac{1}{4}){\ensuremath{\hbar}}^{2}∕(2M{r}^{2})$ is nonvanishing and even attractive for $s$ waves, $m=0$, and the leading near-threshold energy dependence of phase shifts and amplitudes for scattering and quantum reflection is logarithmic in this case. The near-threshold behavior of the $s$-wave phase shifts and amplitudes can nevertheless be characterized by a well-defined scattering length (for potentials falling off faster than $1∕{r}^{2}$) and an effective range (for potentials falling off faster than $1∕{r}^{4}$). For a potential with a bound state at energy ${E}_{n}=\ensuremath{-}{\ensuremath{\hbar}}^{2}{\ensuremath{\kappa}}_{n}^{2}∕(2M)$ very near threshold, the scattering length obeys $a\stackrel{{\ensuremath{\kappa}}_{n}\ensuremath{\rightarrow}0}{\ensuremath{\sim}}2{e}^{\ensuremath{-}{\ensuremath{\gamma}}_{\mathrm{E}}}∕{\ensuremath{\kappa}}_{n}+O({\ensuremath{\kappa}}_{n})$, with no term of order ${\ensuremath{\kappa}}_{n}^{0}$---in contrast to the three-dimensional case. 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 $m\ensuremath{\ne}0$ we give analytical expressions for the near-threshold behavior of phase shifts as well as scattering and quantum reflection amplitudes, which are generally valid, even when $m$ is neither integer nor half-integer.