Stability for quasi-periodically perturbed Hill's equations

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We consider a perturbed Hill's equation of the form ϕ+ (p₀(t) + εp₁(t)) ϕ= 0, where p₀ is real analytic and periodic, p₁ is real analytic and quasi-periodic and is a ``small'' real parameter. Assuming Diophantine conditions on the frequencies of the decoupled system, i.e. the frequencies of the external potentials p₀ and p₁ and the proper frequency of the unperturbed (ε=0) Hill's equation, but without making non-degeneracy assumptions on the perturbing potential p₁, we prove that quasi-periodic solutions of the unperturbed equation can be continued into quasi-periodic solutions if ε lies in a Cantor set of relatively large measure in [-ε₀,ε₀], where ε₀ is small enough. Our method is based on a resummation procedure of a formal Lindstedt series obtained as a solution of a generalized Riccati equation associated to Hill's problem.