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Eigenvalue interlacing for first order differential systems with periodic \(2 \times 2\) matrix potentials and quasi-periodic boundary conditions. (English) Zbl 1409.34075
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.
MSC:
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
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