Geodesic
Scattering problem for the Schr\"odinger equation in the case of a~potential linear in time and coordinate.~I. Asymptotics in the shadow zone
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 14, Tome 140 (1984), pp. 6-17

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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. 
@article{ZNSL_1984_140_a0,
     author = {V. M. Babich and V. P. Smyshlyaev},
     title = {Scattering problem for the {Schr\"odinger} equation in the case of a~potential linear in time and {coordinate.~I.} {Asymptotics} in the shadow zone},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {6--17},
     publisher = {mathdoc},
     volume = {140},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_140_a0/}
}
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V. M. Babich; V. P. Smyshlyaev. Scattering problem for the Schr\"odinger equation in the case of a~potential linear in time and coordinate.~I. Asymptotics in the shadow zone. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 14, Tome 140 (1984), pp. 6-17. http://geodesic.mathdoc.fr/item/ZNSL_1984_140_a0/