an:06980062
Zbl 1410.34262
Currie, Sonja; Roth, Thomas T.; Watson, Bruce A.
Inverse problems for first-order differential systems with periodic \( 2\times 2\) matrix potentials and quasi-periodic boundary conditions
EN
Math. Methods Appl. Sci. 41, No. 15, 5985-5988 (2018).
00415955
2018
j
34L40 34B05 34A55
Dirac system; quasi-periodic boundary conditions; inverse problem
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 \(q=0\).
Jonathan Eckhardt (Loughborough)