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Structure and coordinate dependence of the reflected wave, as well asboundary conditions for quasi-particles of graphene and the two dimensionalelectron gas in sheets with abrupt lattice edges are obtained and analyzed bythe Green's function technique. In particular, the reflection wave functioncontains terms inversely proportional to the distance to the graphene latticeedge. The Dirac equation and the momentum dependence of the wave functions ofthe quasi-particles near the conical points are also found by the perturbationtheory with degeneracy in terms of the Bloch functions taken at the degeneracypoints. The developed approach allows to formulated the validity criteria forthe Dirac equation in a rather simple way.

Scattering problems and boundary conditions for 2D electron gas and graphene