Boundary condition and reflection anomaly in $2+1$ dimensions

It is known that the $2+1$d single Majorana fermion theory has an anomaly of the reflection and it is cancelled when we combine 16 copies of the theory. Therefore, it is expected that the reflection symmetric boundary condition is impossible for one Majorana fermion, but possible for 16 Majorana fermions. In this paper, we consider a reflection symmetric boundary condition, which varies at a point, and find that there is a problem for one Majorana fermion. The problem is the absence of the corresponding outgoing wave to a specific incoming wave into the boundary, which leads the non-conservation of the energy. When we introduce 16 Majorana fermions, we can additionally impose a condition at the point where the boundary condition varies so that there is an outgoing wave corresponding to each incoming wave while preserving the reflection symmetry. In addition, we discuss the connection with the fermion-monopole scattering in $3+1$ dimensions.