Anomalous Threshold Laws in Quantum Sticking

It has been stated that for a short-ranged surface interaction, the probability of a low-energy particle sticking to a surface always vanishes as $s\sim k$ with $k\to 0$ where $k=\sqrt{E}$. Deviations from this so-called universal threshold law are derived using a linear model of particle-surface scattering. The Fredholm theory of integral equations is used to find the global conditions necessary for a convergent solution. The exceptional case of a zero-energy resonance is considered in detail. Anomalous threshold laws, where $s\sim k^{1+\alpha}, \alpha > 0$ as $k\to 0$, are shown to arise from a soft gap in the weighted density of states of excitations; $\alpha$ is determined by the behavior of the weighted density of states near the binding energy.