Zero energy mode for an electron in graphene in a perpendicular magnetic field with constant asymptotics

We study the influence of a perpendicular magnetic field with the asymptotics $B(r\to \infty)= B_0$ in a electrons in graphene. It is shown that the zero-energy solutions can exist only for one pseudospin direction, depending on the sign of the magnetic field in the infinite boundary. This, shows that zero-energy level is robust with respect to possible inhomogeneities of the magnetic field. In addition, we show that the number of the states with zero energy for one pseudospin projection is infinity. These results should be useful in the study of ripples which cause a scattering of Dirac particles in slowly decreasing magnetic fields where the asymptotic states is easy to define.