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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.
Original language | English |
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Pages (from-to) | 369-411 |
Number of pages | 43 |
Journal | Journal of Differential Equations |
Volume | 296 |
DOIs | |
State | Published - 25 Sep 2021 |
ID: 86154239