Asymptotic solutions of whispering-gallery type for the time-dependent Schrödinger equation

Formal asymptotic solutions of the time-dependent Schrödinger equation that are concentrated in the neighborhood of the boundary of a cylindrical region and satisfy the zero-value boundary condition are constructed. It is shown that the leading term in the asymptotic solution can be calculated if the solution is known to the Cauchy problem for the Hamilton system corresponding to the problem of classical mechanics of the motion of a material point subject to a holonomic constraint.