Periodic Dirac operator with dislocation

We consider a Dirac operator with a dislocation potential on the real line. The dislocation potential is a fixed periodic potential on the negative half-line and the same potential but shifted by real parameter $t$ on the positive half-line. Its spectrum has an absolutely continuous part (the union of bands separated by gaps) plus at most two eigenvalues in each non-empty gap. Its resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface. We prove that it has only two simple poles on each open gap: on the first sheet (an eigenvalue) or on the second sheet (a resonance). These poles are called states and there are no other poles. We prove: 1) each state is a continuous function of $t$, and we obtain its local asymptotic; 2) for each $t$ states in the gap are distinct; 3) in general, a state is non-monotone function of $t$ but it can be monotone for specific potentials; 4) we construct examples of operators, which have: a) one eigenvalue and one resonance in any finite number of gaps; b) two eigenvalues or two resonances in any finite number of gaps; c) two static virtual states in one gap.