Geodesic
On the scattering problem for the Schrodinger equation in case of the potential linear in time and coordinate.~II Correctness, smoothness, solution's behaviour in infinity
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 15, Tome 148 (1985), pp. 13-29

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She scattering problem of whispering gallery waves in a vicinity of a boundary flex point is investigated. Theorems of existence, uniqueness,smoothness of the solution, the validity of formal asymptotics when $t\to-\infty$ are proved. 
@article{ZNSL_1985_148_a1,
     author = {V. M. Babich and V. P. Smyshlyaev},
     title = {On the scattering problem for the {Schrodinger} equation in case of the potential linear in time and {coordinate.~II} {Correctness,} smoothness, solution's behaviour in infinity},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {13--29},
     publisher = {mathdoc},
     volume = {148},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_148_a1/}
}
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V. M. Babich; V. P. Smyshlyaev. On the scattering problem for the Schrodinger equation in case of the potential linear in time and coordinate.~II Correctness, smoothness, solution's behaviour in infinity. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 15, Tome 148 (1985), pp. 13-29. http://geodesic.mathdoc.fr/item/ZNSL_1985_148_a1/