Stability for Quasi-Periodically Perturbed Hill's Equations

We consider a perturbed Hill's equation of the form +(p₀(t)+ɛp₁(t))ϕ=0, where p₀ is real analytic and periodic, p₁ is real analytic and quasi-periodic and ɛ ∈ℝ is ``small'. 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 (ɛ=0) Hill's equation, but without making any assumptions on the perturbing potential p₁ other than analyticity, we prove that quasi-periodic solutions of the unperturbed equation can be continued into quasi-periodic solutions if ɛ lies in a Cantor set of relatively large measure in where ɛ₀ 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.