The problem of integrating the Camassa-Holm equation leads to the scattering and inverse scattering problem for the Sturm-Liouville equation -u" + 1\4 u = λwu, where w is a weight function which may change sign but where the left-hand side gives rise to a positive quadratic form so that one is led to a left-definite spectral problem. In this paper the spectral theory and a generalized Fourier transform associated with the equation -u" + 1\4u = λwu posed on a half-line are investigated. An inverse spectral theorem and an inverse scattering theorem are established. A crucial ingredient of the proofs of these results is a theorem of Paley-Wiener type which is shown to hold true. Additionally, the accumulation properties of eigenvalues are investigated. © 2009 Society for Industrial and Applied Mathematics.