Abstract. For the radial Schr6dinger equation with a potential q(x) decreasing at infinity as qo x-', ee (O, 2), the low energy asymptotics of spectral and scattering data is found. In particular, it is shown that for qo > 0 the spectral function vanishes exponentially as the energy k z tends to zero. On the contrary, there is always a zero-energy resonance for q0 < 0. These results determine the local asymptotics of solutions of the time-dependent Schr6dinger equation for large times t. Specifically, for positive potentials its solutions decay as exp (-Oot(Z-~)/(z+~), Oo>O,t~oo. In the case ~s(1,2) it is shown that for -q o > 0 the phase shift tends to +_ oo as k--,0 and its asymptotics is evaluated.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.