The inverse resonance problem for left-definite Sturm-Liouville operators

We address inverse spectral and scattering problems for the half-line, left-definite Sturm-Liouville equation, −u″+qu=λwu. These problems have been considered recently as they are critical in integrating the Camassa-Holm equation. Previous results required that the support of w be free of gaps, relatively open intervals on which w=0 almost everywhere. We relax this condition and prove an inverse spectral theorem that tells to what extent the Weyl-Titchmarsh m-function or the spectral measure determines the coefficients of the equation. Note that, unlike the Schrödinger equation, knowing the spectral measure is not the same as knowing the m-function. We also prove an inverse resonance theorem that explains to what extent the eigenvalues and resonances determine the spectral measure (and, thus, the coefficients). Again, unlike the Schrödinger case, these data are not sufficient; the presence of w multiplying the spectral parameter complicates the analysis. However, we show that, in most cases, only one other number is needed to fully recover the spectral measure.