Efficient semiclassical approximation for bound states in graphene in magnetic field with a small trigonal warping correction

This paper is devoted to the construction of semiclassical spectrum and efficient (simple to implement) explicit semiclassical asymptotic eigenfunctions of the Dirac operator for high-energy bound states in graphene in magnetic field, with considering the effect of trigonal warping [1], [2] to be small. It turns out that the asymptotic spectrum of the operator is invariant under such a perturbation, but due to the symmetry of the problem only, rather than the smallness of this correction.
However, the behavior of asymptotic eigenfunctions is quite different; they are significantly affected by trigonal warping that leads to the breaking of certain symmetries. Density plots of asymptotic eigenfunctions can indicate what should be observed using a scanning tunneling microscope. Our approach to constructing asymptotic solutions is based on developments of previous papers [3]–[6], which present a new method for constructing the solution, simplifying practical application. We also provide a rigorous estimate for the tails of the Fourier series of the amplitudes, which permits one to exclude them from consideration.