Periodic Dirac operator with dislocation
We consider Dirac operators with dislocation potentials on the line. The dislocation potential is a periodic potential for x < 0 and the same potential but shifted by t ∈ R for x > 0. Its spectrum has an absolutely continuous part (the union of bands separated by gaps) plus at most two eigenvalues in each 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: eigenvalues or resonances. These poles are called states and there are no other poles. We prove: 1) states are continuous functions of t, and we obtain their local asymptotics; 2) for each t states in the gap are distinct; 3) states can be monotone or non-monotone functions of t; 4) we construct examples of operators with different types of states in gaps.