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First-order systems in \(\mathbb{C}^2\) on \(\mathbb R\) with periodic matrix potentials and vanishing instability intervals. (English) Zbl 1344.34034
This article is concerned with two-dimensional first order systems of the form \[ JY' + QY = \lambda Y \] on the line, where \(Q\) is a locally integrable real symmetric matrix-valued potential function with period \(\pi\). After studying the corresponding Floquet discriminant, properties of periodic, antiperiodic and auxiliary eigenvalues as well as large \(\lambda\) asymptotics for fundamental systems of solutions, the authors prove their main result. They show that (under the additional assumption that \(Q\) is locally absolutely continuous) all instability intervals vanish if and only if \(Q\) is of the form \(pI\) for some scalar function \(p\), where \(I\) denotes the identity matrix.

MSC:
34B05 Linear boundary value problems for ordinary differential equations
34A55 Inverse problems involving ordinary differential equations
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34L05 General spectral theory of ordinary differential operators
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