Inverse spectral theory for Sturm-Liouville operators with distributional potentials

We discuss inverse spectral theory for singular differential operators on arbitrary intervals (a, b) ⊆ R associated with rather general differential expressions of the type τf = 1 r ( − ( p[f ′ + sf ] )′ + sp[f ′ + sf ] + qf ) , where the coefficients p, q, r, s are Lebesgue measurable on (a, b) with p−1, q, r, s ∈ Lloc((a, b); dx) and real-valued with p 6= 0 and r > 0 almost everywhere on (a, b). In particular, we explicitly permit certain distributional potential coefficients. The inverse spectral theory results derived in this paper include those implied by the spectral measure, by two-spectra and three-spectra, as well as local Borg–Marchenko-type inverse spectral results. The special cases of Schrödinger operators with distributional potentials and Sturm–Liouville operators in impedance form are isolated, in particular.