The paper deals with the spectrum of the Hill operator \({\mathcal L}= -{d^ 2 \over dx^ 2} +q(x)\). It is well known that this spectrum coincides with the union over \(t \in [0, \pi]\) of all solutions \(\mu\) of the equation \(u_ +(\sqrt \mu)=\cos t\), \(0 \leq t \leq \pi\), where \(u_ + (\lambda)\) is the Lyapunov function of the operator \({\mathcal L}\). The author gives, among others, the necessary and sufficient conditions in order that a function \(u_ + (\lambda)\) be the Lyapunov function of \({\mathcal L}\) and also the necessary and sufficient conditions in order that the potential \(q(x)\) be infinitely differentiable on the axis \(R\) or be analytic on \(R\).