Sharp spectral transition for embedded eigenvalues of perturbed periodic Dirac operators

We consider the Dirac equation on L²(R)⊕ L²(R) {align} Ly= {pmatrix} 0&-1 1&0 {pmatrix} {pmatrix} y_1 y_2 {pmatrix}'+ {pmatrix} p&q q&-p {pmatrix}{pmatrix} y_1 y_2 {pmatrix}+ V{pmatrix} y_1 y_2 {pmatrix}=λ y, {align} where y=y(x,λ)=y₁(x,λ)y₂(x,λ), p and q are real $1$-periodic, and {align} V={pmatrix} V(x)&0 0&-V(x) {pmatrix} {align} is the perturbation which satisfies $V(x)=o(1)$ as x→∞. Under such perturbation, the essential spectrum of L coincides with that there is no perturbation. We prove that if V(x)=o(1)1+x as x→∞ or x→-∞, 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)=O(1)1+x as x→∞ 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)<h(x)1+x as x→∞ so that the set becomes embedded eigenvalues, where $h(x)$ is any given function with limx→±∞h(x)=∞.