A variational-integral perturbation method (VIPM) is established by combining the variational perturbation with the integral perturbation. The first-order corrected wave functions are constructed, and the second-order energy corrections for the ground state and several lower excited states are calculated by applying the VIPM to the hydrogen atom in a strong uniform magnetic field. Our calculations demonstrated that the energy calculated by the VIPM only shows a negative value, which indicates that the VIPM method is more accurate than the other methods. Our study indicated that the VIPM can not only increase the accuracy of the results but also keep the convergence of the wave functions.

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The extended line defect of graphene is an extraordinary candidate in valleytronics while the high valley polarization can only occur for electrons with high incidence angles which brings about tremendous challenges to experimental realization. In this paper, we propose a simple scheme to filter one conical valley state by applying a local magnetic field. It is found that due to the movement of the Dirac points, the transmission profiles of the two valleys are shifted along the injection-angle axis at the same pace, resulting in the peak transmission of one valley state being reduced drastically while remaining unaffected for the other valley state, which induces nearly perfect valley polarization. This scheme can be easily implemented in experiments and plays a key role in graphene valleytronics.