A gap in the energy spectrum of the one-dimensional Dirac operator

One considers the one-dimensional Dirac operator with a slowly oscillating potential
\begin{equation} H=\begin{pmatrix} 0 & 1\\ -1 &0 \end{pmatrix}\dfrac{d}{dx}+q\begin{pmatrix} \cos Z(x) & \sin Z(x)\\ \sin Z(x) & -\cos Z(x)\end{pmatrix},\quad x\in(-\infty,\infty),\quad q-\mathrm{const}, \end{equation}
where $Z(x)\in C^1(\mathbf R^1)$ and $Z(x)\underset{x\to\pm\infty}\to C\pm|x|^\alpha$, $0<\alpha<1$, $C\pm-\mathrm{const}$. The following statement holds. The double absolutely continuous spectrum of the operator (1) fills the intervals $(-\inftu,-|q|)$, $(|q|,\infty)$. The interval $(-|q|,|q|)$ is free from spectrum. The operator has a simple eigenvalue only for $\operatorname{sign}C_+=\operatorname{sign}C_-$, situated either at the point (under the condition $C_+>0$) or at the point $\lambda=-|q|$ (under the condition). The proof is based on the investigation of the coordinate asytnptotics of the corresponding equation.