Sharp spectral transition for embedded eigenvalues of perturbed periodic Dirac operators

We consider the Dirac equation on $L^2(\mathbb{R})\oplus L^2(\mathbb{R})$ \begin{align} Ly= \begin{pmatrix} 0&-1 1&0 \end{pmatrix} \begin{pmatrix} y_1 y_2 \end{pmatrix}'+ \begin{pmatrix} p&q q&-p \end{pmatrix}\begin{pmatrix} y_1 y_2 \end{pmatrix}+ V\begin{pmatrix} y_1 y_2 \end{pmatrix}=λy,\nonumber \end{align} where $y=y(x,\lambda )=\left(\genfrac{}{}{0px}{}{y_1(x,\lambda )}{y_2(x,\lambda )}\right)$, $p$ and $q$ are real $1$-periodic, and \begin{align} V=\begin{pmatrix} V(x)&0 0&-V(x) \end{pmatrix}\nonumber \end{align} is the perturbation which satisfies $V(x)=o(1)$ as $\text{abs}x\to \mathrm{\infty }\mathrm{.}$ Under such perturbation, the essential spectrum of $L$ coincides with that there is no perturbation. We prove that if $V(x)=\frac{o(1)}{1+\text{abs}x}$ as $x\to \mathrm{\infty }$ or $x\to -\mathrm{\infty }$, then there is no embedded eigenvalues (eigenvalues appear in the essential spectrum). For any given finite set inside of the essential spectrum which satisfies the non-resonance assumption, we construct smooth potentials with $V(x)=\frac{O(1)}{1+\text{abs}x}$ as $\text{abs}x\to \mathrm{\infty }$ so that the set becomes embedded eigenvalues. For any given countable set inside of the essential spectrum which satisfies the non-resonance assumption, we construct smooth potentials with $V(x)<\frac{\text{abs}h(x)}{1+\text{abs}x}$ as $\text{abs}x\to \mathrm{\infty }$ so that the set becomes embedded eigenvalues, where $h(x)$ is any given function with ${\lim }_{x\to \pm \mathrm{\infty }}\text{abs}h(x)=\mathrm{\infty }\mathrm{.}$