ИСТИНА

We propose constructive semi-classical asymptotics for the 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 with low mass. The problem is reduced by standard semi-classical methods to a pencil of magnetic Schr¨odinger operators with a correction. In both cases, the classical system defined by the main symbol turns out to be integrable, but the correction destroys the integrability. In case (a), where the correction removes the frequency degeneracy (resonance), using the averaging method, we 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 main symbol is nondegenerate. Fixing an invariant torus with Diophantine frequencies for this system and finding 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 rule and lie in a small neighborhood of the chosen Diophantine torus. In both cases, the construction of the asymptotics of the eigenfunctions is based on the global representation in terms of the Airy function and its derivative for the Maslov’s canonical operator on a two-dimensional torus projected onto the configuration space into an annular domain with two simple caustics. We also give some numerical examples that illustrate that the obtained formulas are efficient. The work was supported by the Russian Science Foundation (project No. 21-11-00341).