Inverse scattering as an infinite period limit

Abstract

The purpose of this paper is to show how the well-known Dyson formula giving the solution of the KdV equation (or equivalently the solution of the inverse scattering problem for the Schrödinger operator) in the case of initial data with compact support, can be derived as an infinite period limit of the Its-Matveev formula which gives all periodic (and more) solutions of KdV (or. equivalently, solves the inverse Floquet-spectral problem) in terms of theta functions. This problem is motivated by the natural question of how to extend the known theory of KdV to cases other than those of decaying initial data on the one hand, and the algebro-geometric case on the other hand. We solve our problem by considering a local smooth potential which is periodized and letting the period go to infinity (at the level of the spectral problem). We use the Riemann-Hilbert formulation of the inverse spectral problem for the periodic case, and show how the Riemann-Hilbert problem corresponding to the inverse scattering theory for the initial local potential is recovered, as the period tends to infinity.