The inverse periodic problem for Hill's equation with a finite-gap potential

Abstract

Let H(q) = d*/d~’ + q(x) be the Hill’s operator with a periodic potential q of period one. Consider the following inverse problem. Find all potentials q(x) with the same periodic spectrum as a given potential p(x). Two explicit solutions are known. For finite-gap potentials, Its and Matveev [6]. give an explicit formula in terms of theta functions. Their solution was generalized by McKean and Trubowitz [lo] to infinite-gap potentials. Finkel et al. [5] give a second formula which expresses q in terms of the base potential p, the Dirichlet eigenfunctions of p and the Floquet solutions of p. The formula involves no algebraic geometry. In this paper we represent finite-gap potentials q as rational functions of the base potential p and the derivatives ofp. The representation makes use of the Finkel-Isaacson-Trubowitz (FIT) formula.