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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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},
     year = {1985},
     volume = {148},
     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/