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

Abstract: This paper is devoted to the construction of semiclassical spectrum and efficient (simple to implement) explicit semiclassical asymptotic eigenfunctions of the Dirac operator for relatively high-energy bound states in graphene in magnetic field, considering the effect of trigonal warping [11, 16] to be small. It turns out that the asymptotic spectrum of the operator remains unchanged under such a perturbation due to the symmetry of the problem 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 might be observed using a scanning tunneling microscope. Our approach to constructing asymptotic solutions is based on developments of works [9, 1, 5], which present a new method for constructing the solution, simplifying practical application.