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In this paper, we consider the reducibility of the quasi-periodic linearHamiltonian system $$\dot{x}=(A+\varepsilon Q(t))x, $$ where $A$ is a constantmatrix with possible multiple eigenvalues, $Q(t)$ is analytic quasi-periodicwith respect to $t$, and $\varepsilon$ is a sufficiently small parameter. Undersome non-resonant conditions, it is proved that, for most sufficiently small$\varepsilon$, the Hamiltonian system can be reduced to a constant coefficientHamiltonian system by means of a quasi-periodic symplectic change of variableswith the same basic frequencies as $Q(t)$. Application to quasi-periodic Hill'sequation is also given.

The reducibility of quasi-periodic linear Hamiltonian systems and its application to Hill's equation