Dubrovin equation for periodic Dirac operator on the half-line

We consider the Dirac operator with a periodic potential on the half-line with the Dirichlet boundary condition at zero. Its spectrum consists of an absolutely continuous part plus at most one eigenvalue in each open gap. The resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface with a unique simple pole on each open gap: on the first sheet (an eigenvalue) or on the second sheet (a resonance). These poles are called levels and there are no other poles. If the potential is shifted by real parameter t, then the continuous spectrum does not change but the levels can change their positions. We prove that each level is smooth and in general, non-monotonic function of t. We prove that a level is a strictly monotone function of t for a specific potential. Using these results, we obtain formulas to recover potentials of special forms.

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