Geodesic
Examples of exactly solvable scattering problems for the parabolic equation of diffraction theory
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 9, Tome 78 (1978), pp. 184-202

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Examples of exactly solvable scattering problem are considered for equations of Schrodinger type with a potential having the same character of behavior as in the case of the problem of whispering gallery waves in a neighborhood of an inflection point of the boundary. The qualitative behavior of the exact solutions found is investigated, and it is compared with the numerical solution of the aforementioned problems regarding whispering gallery waves. 
@article{ZNSL_1978_78_a13,
     author = {M. M. Popov},
     title = {Examples of exactly solvable scattering problems for the parabolic equation of diffraction theory},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {184--202},
     publisher = {mathdoc},
     volume = {78},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1978_78_a13/}
}
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M. M. Popov. Examples of exactly solvable scattering problems for the parabolic equation of diffraction theory. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 9, Tome 78 (1978), pp. 184-202. http://geodesic.mathdoc.fr/item/ZNSL_1978_78_a13/