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 y₁ y₂ pmatrix'+ pmatrix p&q q&-p pmatrixpmatrix y₁ y₂ pmatrix+ Vpmatrix y₁ y₂ pmatrix= y, align where y=y (x, ) =y₁ (x, ) y₂ (x, ), 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 ₗh (x) =.