Fermion threshold energy states

In this paper, the authors show that for a three-dinemsional non-relativistic spinor confined on a plane, thje spin-up component obeys the same equation of motion as a two-dimensional spinor. Threshold energy solution is investigated when the electron is moving in the vortex field. It can be proved from the index theorem that the existence of the threshold states depends on the magnetic flux only, not on the magnetic field configuration