Second order asymptotics for Krein indefinite multipliers with multiplicity two

Abstract

We consider linear Hamiltonian equations in ${\mathbb{R}}^4$ of the following type \begin{equation} \frac{\mathrm{d}γ}{\mathrm{d}t}(t)=J_{4}A(t)γ(t), γ(0)\in\operatorname{Sp}(4,\mathbb{R}), \end{equation} where $J=J_4\stackrel{\text{def}}{=}\left[\begin{array}{cc}0& {\mathrm{Id}}_2\\ -{\mathrm{Id}}_2& 0\end{array}\right]$ and $A:t\mapsto A(t)$ is a $C^1$-continuous curve in the space of $4\times 4$ real matrices which are symmetric. We obtain second order asymptotics for the eigenvalues bifurcated from non-real Krein indefinite eigenvalues with multiplicity two.