Quadratic operator pencils associated with the conservative Camassa-Holm flow

We discuss direct and inverse spectral theory for a Sturm-Liouville type problem with a quadratic dependence on the eigenvalue parameter, -f'' +1/4f = z ωf + z²vf, which arises as the isospectral problem for the conservative Camassa-Holm flow. In order to be able to treat rather irregular coefficients (that is, when ω is a real-valued Borel measure on ℝ and v is a non-negative Borel measure on ℝ), we employ a novel approach to study this spectral problem. In particular, we provide basic self-adjointness results for realizations in suitable Hilbert spaces, develop (singular) Weyl-Titchmarsh theory and prove several basic inverse uniqueness theorems for this spectral problem.