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 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"+14u=λwu. © 2012 Elsevier Inc.