Amplitudes of multichannel scattering on the barrier with a constant height in the scattering direction

The immersing method is applied to solve the N-channel scattering problem. In particular, we consider the particle scattering on a multidimensional potential barrier, which is constant in the scattering direction and arbitrary in the lateral direction. For this case the scattering amplitudes t m and r m (m = 1, 2, …, N) are determined. Transition from the obtained formulas to the case of thin potential is performed. For this case analytical expressions of transmission amplitudes t m and reflection amplitudes r m are obtained. We show that the product of transmission and reflection amplitudes in the channel m does not depend on the scattering channel. It is assumed that the scattering particle falls on the potential with the longitudinal wave vector k l corresponding to the channel l.