Stability for quasi-periodically perturbed Hill’s equations

We consider a perturbed Hill’s equation of the form φ̈ + (p0(t) + εp1(t))φ = 0, where p0 is real analytic and periodic, p1 is real analytic and quasi-periodic and ε ∈ 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 (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] ⊂ R, 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.