Resonance tunneling through a three-dimensional asymmetric potential with a barrier

Resonance tunneling through a three-dimensional axial-symmetrical potential with a barrier, for which the separation of variables in the Schroedinger equation is impossible, is considered in the semiclassical approximation. Rules are obtained for joining the wave functions which generalize the known one-dimensional rules by Jeffries. A contribution to the flux (the scattering amplitude) from states localized in a vicinity of the potential symmetry axis is calculated. With increase of the state transverse energy the contribution falls exponentially