Exceptional solutions of Hill equations (Q1346231)

The paper includes 5 sections. In the introduction a short review of the previously published results about the solutions of the Hill equations is made. In Section 2 are considered two solutions \(u_ 1 (t)\) and \(u_ 2 (t)\) of the differential equation \(y'' = q(t)y\), \(q \in C^ 0(R)\), \(q(t + \pi) = q(t)\), satisfying the initial conditions \(u_ 1(0) = 0\), \(u_ 1'(0) = 1\), \(u_ 2 (0) = 1\), \(u_ 2'(0) = 0\). Let \(d = u_ 1'(\pi) + u_ 2 (\pi)\). Four cases, according to the values of \(d\) are distinguished and in this way a global classification of Hill equations is presented summarizing the different results on these equations. The connection between the Lyapunov exponent and the rotation number of the dynamical system induced by the Hill equation is given. In Section 3 is formulated the main result of the paper, which includes the list of exceptional solutions for all Hill equations. In Section 4 are given the formulations and the proofs of three lemmas, which give some estimates on integrals connected with periodic functions. In Section 5 the proof of the main result of the paper is presented.