Inverse problems for first-order differential systems with periodic \( 2\times 2\) matrix potentials and quasi-periodic boundary conditions. (English) Zbl 1410.34262

Currie, Sonja; Roth, Thomas T.; Watson, Bruce A.

Math. Methods Appl. Sci. 41, No. 15, 5985-5988 (2018).

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 \(q=0\).

Reviewer: Jonathan Eckhardt (Loughborough)

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

Keywords:

Dirac system; quasi-periodic boundary conditions; inverse problem