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In the paper complete systems of exact solutions for Dirac and Weyl equationsin the Lobachevsky space are constructed on the base of the method ofseparation of the variables in quasi-cartesian coordinates. An extendedhelicity operator is introduced. It is shown that solution constructed whentranslating to the limit of vanishing curvature coincide with common plane wavesolutions on Minkowski space going in opposite z-directions. It is shown theproblem posed in Lobachevsky space simulates a situation in the flat space fora quantum-mechanical particle of spin 1/2 in a 2-dimensional potential barriersmoothly rising to infinity on the right.

Quasi-plane waves for a particle with spin 1/2 on the background of Lobachevsky geometry: simulating of a special medium