an:06905105
Zbl 1409.34075
Currie, Sonja; Roth, Thomas T.; Watson, Bruce A.
Eigenvalue interlacing for first order differential systems with periodic \(2 \times 2\) matrix potentials and quasi-periodic boundary conditions
EN
Oper. Matrices 12, No. 2, 489-499 (2018).
00406867
2018
j
34L40 34L15 47E05
Dirac system; quasi-periodic eigenvalue problems; interlacing
The paper deals with the self-adjoint Dirac system in the form
\[
JY' + Q Y = \lambda Y, \tag{1}
\]
where \(J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\), \(Q = \begin{pmatrix} q_1 & q \\ q & q_2 \end{pmatrix}\), the functions \(q\), \(q_1\) and \(q_2\) are real-valued, integrable and \(\pi\)-periodic. It is shown that the eigenvalues of the boundary value problem for equation (1) with the boundary conditions
\[
Y(\pi) = \pm R(\theta) Y(\theta),
\]
where \(R(\theta) = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{pmatrix}\), coincide with the values of \(\lambda\), such that the discriminant \(\Delta_{\theta} = \text{Tr} (\mathbb Y(\pi)^T R(\theta))\) equals \(\pm 2\). Here \(\mathbb Y(x)\) is the solution of (1), satisfying the initial condition \(\mathbb Y(0) = \mathbb I\). The authors obtain explicit formulas for the \(\lambda\)-derivative of the discriminant \(\Delta_{\theta}\) and monotonicity results for the first and the second \(\lambda\)-derivatives. The main results of the paper are several interlacing theorems for the eigenvalues of (1), corresponding to various boundary conditions.
Natalia Bondarenko (Saratov)