ИСТИНА

We study the stationary 2-D Dirac equation for graphene with an axially symmetric potential, constant magnetic field and variable mass (impurity). The mass has no a xial symmetry, and it plays the role of a perturbation. We are interested in (formal) semi-classica l asymptotics of its solutions or, equivalently, quasi-modes of Dirac operator. The principal symbol of the o perator defines a completely integrable system with a family of invariant Liouville tori. The subpri ncipal symbol (and, hence, the transport equation) turns out to be non-trivial. To solve it one has to i mpose two conditions on a torus: 1) it has to be Diophantine, and 2) it has to satisfy Bohr-Sommerfe ld quantization rule. However, these two condtions may contradict to each other. Applying ideas b y Lazutkin, Dobrokhotov, Rouleux and others, we show how to evade this condtradiction, and presen t an efficient algorithm for constructing quasi-modes. This is a joint work with S. Yu. Dobrokhotov and S. B. Kuksin