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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.