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

This work develops a scattering and an inverse scattering theory for the Sturm-Liouville equation u '' qu = lambda wu 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 left-definite 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 Paley-Wiener theorem linking growth properties of transforms with support properties of functions. One motivation for this investigation comes from the Camassa-Holm equation for which the solution of the Cauchy problem can be achieved by the inverse scattering transform for -u '' + 1/4 u = lambda wu. (c) 2012 Elsevier Inc. All rights reserved