Abstract:
The formal asymptotics of the scattering problem for the Schrödinger equation with a linear potential as $x+|t|\to+\infty$ is studied. In the shadow zone a formal asymptotic expansion is constructed which is matched with the known asymptotics as $t\to-\infty$. The expansion constructed loses asymptotic character near the curve $x=\frac16t^3$ (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.
Citation:
V. M. Babich, V. P. Smyshlyaev, “Scattering problem for the Schrödinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone”, Mathematical problems in the theory of wave propagation. Part 14, Zap. Nauchn. Sem. LOMI, 140, "Nauka", Leningrad. Otdel., Leningrad, 1984, 6–17; J. Soviet Math., 32:2 (1986), 103–112
\Bibitem{BabSmy84}
\by V.~M.~Babich, V.~P.~Smyshlyaev
\paper Scattering problem for the Schr\"odinger equation in the case of a~potential linear in time and coordinate.~I. Asymptotics in the shadow zone
\inbook Mathematical problems in the theory of wave propagation. Part~14
\serial Zap. Nauchn. Sem. LOMI
\yr 1984
\vol 140
\pages 6--17
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl4065}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=765714}
\zmath{https://zbmath.org/?q=an:0557.35024}
\transl
\jour J. Soviet Math.
\yr 1986
\vol 32
\issue 2
\pages 103--112
\crossref{https://doi.org/10.1007/BF01084146}
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V. S. Buldyrev, A. M. Vershik, I. A. Ibragimov, A. M. Il'in, A. P. Kiselev, L. D. Faddeev, “Vasilii Mikhailovich Babich (on his 70th birthday)”, Russian Math. Surveys, 57:3 (2002), 627–635
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