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.
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Communicated by J. Ginibre
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Yafaev, D.R. The low energy scattering for slowly decreasing potentials. Commun.Math. Phys. 85, 177–196 (1982). https://doi.org/10.1007/BF01254456
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DOI: https://doi.org/10.1007/BF01254456