Scattering and inverse scattering for a left-definite Sturm–Liouville problem

Abstract

This work develops a scattering and an inverse scattering theory for the Sturm-Liouville equation −u + qu = λwu where w may change sign but q 0. Thus the left-hand side of the equation gives rise to a positive quadratic form and one is led to a leftdefinite spectral problem. The crucial ingredient of the approach is a generalized transform built on the Jost solutions of the problem and hence termed the Jost transform and the associated PaleyWiener theorem linking growth properties of transforms with support properties of functions. One motivation for this investigation comes from the CamassaHolm equation for which the solution of the Cauchy problem can be achieved by the inverse scattering transform for −u + 1 4 u = λwu.

A uniqueness result for one‐dimensional inverse scattering

“…It seems very general uniqueness theorems for inverse scattering may be deduced from this. We refer to [7].…”

The Spectral Problem for the Dispersionless Camassa–Holm Equation

“…It should be noted that the methods of this note combined with those of [3] allow one to prove a uniqueness theorem for inverse scattering in the case κ = 0 for the case when w is a measure, extending the results of [3] where it was assumed that w ∈ L 1 loc . These results do not appear to be accessible using de Branges' theory.…”

“…For the inverse theory Eckhardt [11] uses de Branges' theory of Hilbert spaces of entire functions. Our approach is different and analogous to that in our paper [3].…”

“…In [3] we discussed scattering and inverse scattering in the case κ = 0, which is the important case for shallow water waves. We did not discuss the zero dispersion case κ = 0, which is relevant in some other situations, but this case was treated by Eckhardt and Teschl in [10], based on the results of Eckhardt [11].…”

Quadratic operator pencils associated with the conservative Camassa-Holm flow

“…Since the coefficient ω is allowed to change sign and because of the presence of the measure υ, spectral theory for (1.1) is outside of most standard theory for Sturm-Liouville problems and requires distinct methods to deal with it. In particular, direct and inverse spectral theory for (1.1) is still not sufficiently developed for applications to the Camassa-Holm flow (but see [3,5,6,7,12,14,21,22,23,27]). Moreover, except for [22], all of these references only deal with the case when the measure υ is not present at all.…”