The low energy scattering for slowly decreasing potentials

Abstract

For the radial Schrödinger equation with a potentialq(x) decreasing at infinity asq ₀ q ^−α, α∈(0, 2), the low energy asymptotics of spectral and scattering data is found. In particular, it is shown that forq ₀>0 the spectral function vanishes exponentially as the energyk ² tends to zero. On the contrary, there is always a zero-energy resonance forq ₀<0. These results determine the local asymptotics of solutions of the time-dependent Schrödinger equation for large timest. Specifically, for positive potentials its solutions decay as exp(−ϑ₀ t ^{(2−α)/(2+α)}, ϑ₀>0,t→∞. In the case α∈(1, 2) it is shown that for ±q ₀>0 the phase shift tends to ±∞ ask→0 and its asymptotics is evaluated.