Examples of exactly solvable scattering problems for the parabolic equation of diffraction theory

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.