Second order asymptotics for Krein indefinite multipliers with multiplicity two (Q2208469)

The authors begin with linear Hamiltonian equations on \(\mathbb {R}^{2n}\) of the form \(\frac{d\gamma}{dt} = J_{2n} A(t) \gamma(t)\), where \(\gamma(0)\) is a \(2n \times 2n\) real symplectic matrix, \(J_{2n}\) is the standard \(2n \times 2n\) symplectic matrix, and \(A(t)\) is a \(C^1\) curve in the space of \(2n \times 2n\) real symmetric matrices. Such systems arise naturally from perturbations of linearized Hamiltonian equations. The authors' primary result for this system is as follows. If \(\lambda_0\) is an eigenvalue of \(\gamma(0)\) on the unit circle with \(\lambda_0 \ne \pm 1\), the algebraic multiplicity of \(\lambda_0\) is two and its geometric multiplicity is one, then there are relatively simple expressions for the second order asymptotics of the bifurcated eigenvalues \(\lambda_1(t)\) and \(\lambda_2(t)\) in terms of two complex vectors and the value of \(A(0)\). They also derive an expression for the derivative of the real part of the sum of \(\lambda_1(t)\) and \(\lambda_2(t)\) at \(t = 0\). The results described here are obtained by adapting the argument used in [\textit{Y. Chang} et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 36, No. 1, 75--102 (2019; Zbl 1409.37058)]. A related result is obtained for the perturbed system \(\frac{d\gamma}{dt}(t,\epsilon) = J_{2n} A(t, \epsilon) \gamma(t, \epsilon)\), where \(A(t,\epsilon)\) and its first and second derivatives with respect to \(\epsilon\) are jointly continuous and where \(\gamma(0,\epsilon)\) is the \(2n \times 2n\) identity matrix. Finally, the authors propose a possible application of their results to the linear stability problem of elliptic Lagrangian solutions of the planar three-body problem.