The low energy scattering for slowly decreasing potentials |
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Authors: | D R Yafaev |
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Institution: | (1) Leningrad Department of Mathematical Institute, 27, r. Fontanka, 191011 Leningrad, USSR |
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Abstract: | For the radial Schrödinger equation with a potentialq(x) decreasing at infinity asq
0
q
–, (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(–0
t
(2–)/(2+), 0>0,t. In the case (1, 2) it is shown that for ±q
0>0 the phase shift tends to ± ask0 and its asymptotics is evaluated. |
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Keywords: | |
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