Effective-range theory for quantum reflection amplitudes

Abstract

We formulate an effective-range theory for the near-threshold behavior of the amplitudes describing quantum reflection in attractive potential tails by adapting the effective-range theory of ordinary elastic scattering to the case of incoming boundary conditions at small distances. For homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>5$, the effective range turns out to be a real multiple (with known coefficient) of the complex scattering length which defines the leading, linear momentum dependence of phase and modulus of the quantum reflection amplitude. Analytical expressions are also given for the leading and next-to-leading terms in the near-threshold behavior of the quantum reflection amplitudes for homogeneous attractive potentials proportional to $\ensuremath{-}1∕{r}^{3}$, $\ensuremath{-}1∕{r}^{4}$, and $\ensuremath{-}1∕{r}^{5}$.