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Inverse problems for first-order differential systems with periodic $$2\times 2$$ matrix potentials and quasi-periodic boundary conditions. (English) Zbl 1410.34262
This short article is concerned with the first order system $JY' + QY= \lambda Y$ on the interval $$[0,\pi]$$, where $J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \qquad Q = \begin{pmatrix} q_1 & q \\ q & q_2 \end{pmatrix},$ and $$q$$, $$q_1$$, $$q_2$$ are real-valued and integrable functions on $$(0,\pi)$$. For some fixed $$\theta\in[0,\pi]$$, the coupled boundary conditions $Y(\pi) = \pm R(\theta)Y(0)$ are imposed, where $$R(\theta)$$ is the matrix given by $R(\theta) = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}.$ It is shown that all eigenvalues of the corresponding two boundary value problems are double if and only if $$q_1=q_2$$ and $$q=0$$.
##### MSC:
 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34B05 Linear boundary value problems for ordinary differential equations 34A55 Inverse problems involving ordinary differential equations
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