A note on a linear spectral theorem for a class of first order systems in $R^{2N}$

Abstract

Along the lines of Atkinson, a spectral theorem is proved for the boundary value problem $$ \left\{\begin{array}{l} Jz' + f(t) J z + P(t) z= \lambda B(t) z \\ x(0) = x(T) =0, \\ \end{array}\right. t \in [0, T], z=(x, y) \in \mathbb{R}^N \times \mathbb{R}^N, $$ where $f(t)$ is real-valued and $P(t), B(t)$ are symmetric matrices, with $B(t)$ positive definite. A suitable rotation index associated to the system is used to highlight the connections between the eigenvalues and the nodal properties of the corresponding eigenfunctions.