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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 \(q=0\).
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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