An inverse Sturm-Liouviile problem for a Hill's equation

Résumé

In this paper, we consider Hill's equation -y′′+q(x)y=λy, where q∈L¹[0,π]. A Hill equation defined on a semi-infinite interval will in general have a mixed spectrum. The continuous spectrum will in general consist of an infinite number of disjoint finite intervals. Between these intervals, point eigenvalues can exist. It is shown that under suitable hypotheses on the spectrum a full knowledge of the spectrum leads to a unique determination of the potential function in the Hill's equation. Moreover , it is shown here that if q(x) is prescribed over the interval [(π/2),π], then a single spectrum suffices to determined q(x) on the interval [0,(π/2)].