The low energy scattering for slowly decreasing potentials
Abstract
For the radial Schrödinger equation with a potential q( x) decreasing at infinity as q 0 q -α, α∈(0, 2), the low energy asymptotics of spectral and scattering data is found. In particular, it is shown that for q 0>0 the spectral function vanishes exponentially as the energy k 2 tends to zero. On the contrary, there is always a zero-energy resonance for q 0<0. These results determine the local asymptotics of solutions of the time-dependent Schrödinger equation for large times t. 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 ±∞ as k→0 and its asymptotics is evaluated.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- June 1982
- DOI:
- 10.1007/BF01254456
- Bibcode:
- 1982CMaPh..85..177Y