ИСТИНА

We consider the spectral problem for the operator $\widehat{\mathcal L}_{\rm TW} = {\mathcal L}_{\rm TW}(\widehat p, x)$: $$ \widehat{\mathcal L}_{\rm TW} \Psi = \mathcal E\Psi, $$ where the symbol has the form $$ {\mathcal L}_{\rm TW}(p,x) = \begin{pmatrix} U(x,h)+M(x,h)& {\mathbf{p}}_1 - i{\mathbf{p}}_2 \\ {\mathbf{p}}_1+i{\mathbf{p}}_2& U(x,h)-M(x,h) \end{pmatrix}+\mu \begin{pmatrix} 0 & ({\mathbf{p}}_1+i{\mathbf{p}}_2)^2 \\ ({\mathbf{p}}_1-i{\mathbf{p}}_2)^2 & 0 \end{pmatrix}. $$ This equation describes charge carriers in graphene with impurity mass $M$, placed in an electric potential $U$ and a constant magnetic field. Here, $\mathbf{p}_j = p_j + A_j(x)$, $\widehat{\mathbf{p}}_j = \widehat{p}_j + A_j(x)$, where $A_1 = Bx_2/2$, $A_2 = -Bx_1/2$, $\widehat{p}_j = -ih \partial/\partial x_j$, $\mu =O(h)$, and $h\to 0$. By scalarizing the problem, we construct spectral series $(\mathcal E, \Psi)$, with $\Psi$ expressed in terms of Airy functions, assuming that $U$ and $M$ are such that the principal symbols of the scalar Hamiltonians are integrable.