Abstract
The formal asymptotics of the scattering problem for the Schrödinger equation with a linear potential as x+¦t¦→+∞ is studied. In the shadow zone a formal asymptotic expansion is constructed which is matched with the known asymptotics as t→−∞ The expansion constructed loses asymptotic character near the curve x=1/6 t3 (in the so-called projector zone). An assumption regarding the analogous behavior of the asymptotic series in the projector zone makes it possible to construct an expansion in the post-projection zone which goes over into the formulas for creeping waves.
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Literature cited
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Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 6–17, 1984.
In conclusion, the authors would like to bring to the reader's attention another approach to asymptotics in the projector zone proposed by M. M. Popov (see the present collection).
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Babich, V.M., Smyshlyaev, V.P. Scattering problem for the Schrödinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone. J Math Sci 32, 103–112 (1986). https://doi.org/10.1007/BF01084146
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DOI: https://doi.org/10.1007/BF01084146