Inverse Spectral and Scattering Theory for the Half-Line Left-Definite Sturm-Liouville Problem

together with appropriate boundary conditions, in the space Lw of functions square integrable with respect to the weight w, i.e., the normsquare of the space is ‖u‖2 = ∫ |u|2w. A basic assumption for this to be possible is that w ≥ 0. In some situations of interest this is not the case, but instead one has p > 0, q ≥ 0. One may then use as a norm-square the integral ∫ (p|u′|2 + q|u|2), and a problem of this type is usually called left definite. A left definite problem of current interest is the spectral problem associated with the Camassa-Holm equation, which is of the form