Constructive Semiclassical Asymptotics of Bound States of Graphene in a Constant Magnetic Field with Small Mass

The paper deals with constructive semiclassical asymptotics of eigenfunctions of the Dirac operator describing graphene in a constant magnetic field. Two cases are considered: (a) a strong magnetic field, and (b) a radially symmetric electric field and small mass. Using standard semiclassical methods, we reduce the problem to a pencil of magnetic Schrödinger operators with a correction term. In both cases, the classical system defined by the principal symbol turns out to be integrable, but the correction term destroys the integrability. In case (a), where the correction removes the frequency degeneracy (resonance), we use the averaging method to reduce the problem to an integrable system not only in the leading approximation but also with the correction taken into account. The tori of the resulting system generate a series of asymptotic eigenfunctions of the original operator. In case (b), the system defined by the principal symbol is nondegenerate. Fixing an invariant torus with Diophantine frequencies for this system and looking for a solution of the transport equation for it, we obtain a series of asymptotic eigenfunctions that are in one-to-one correspondence with tori that satisfy the Bohr–Sommerfeld quantization rule and lie in a small neighborhood of the chosen Diophantine torus. In both cases, the construction of the asymptotic eigenfunctions is based on the global representation of the Maslov canonical operator on a two-dimensional torus projected onto the configuration space into an annular domain with two simple caustics via the Airy function and its derivative. A numerical implementation of our formulas in examples shows their efficiency.