We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals \((a,b) \subseteq \mathbb{R}\) associated with rather general differential expressions of the type \begin{equation*}\tau f = \frac{1}{\tau} (-(p[f'+sf])'+sp[f'+sf]+qf),\end{equation*} where the coefficients \(p, q, r, s\) are real-valued and Lebesgue measurable on \((a,b)\), with \(p \neq 0\), \(r > 0\) a.e. on \((a,b)\), and \(p^{-1}, q, r, s \in L_{loc}^1((a,b),dx)\), and \(f\) is supposed to satisfy \begin{equation*} f \in AC_{loc}((a,b)), p[f'+sf] \in AC_{loc}((a,b)). \end{equation*} In particular, this setup implies that \(\tau\) permits a distributional potential coefficient, including potentials in \(H_{loc}^{-1}((a,b))\). We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator \(T_{max}\), or equivalently, all self-adjoint extensions of the minimal operator \(T_{min}\), all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of \(T_{min}\). In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira m-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of \(T_{min}\). Finally, in the special case where \(\tau\) is regular, we characterize the Krein-von Neumann extension of \(T_{min}\) and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups)

Abstract. Motivated by the study of certain nonlinear wave equations (in particular, the Camassa-Holm equation), we introduce a new class of generalized indefinite strings associated with differential equations of the form, υ is a non-negative Borel measure on [0, L) and z is a complex spectral parameter. Apart from developing basic spectral theory for these kinds of spectral problems, our main result is an indefinite analogue of M. G. Krein's celebrated solution of the inverse spectral problem for inhomogeneous vibrating strings.

Abstract. We give a comprehensive treatment of Sturm-Liouville operators whose coefficients are measures including a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl-Titchmarsh-Kodaira theory. We avoid previous technical restrictions and, at the same time, extend all results to a larger class of operators. Our operators include classical SturmLiouville operators, Sturm-Liouville operators with (local and non-local) δ and δ ′ interactions or transmission conditions as well as eigenparameter dependent boundary conditions, Krein string operators, Lax operators arising in the treatment of the Camassa-Holm equation, Jacobi operators, and Sturm-Liouville operators on time scales as special cases.

Abstract. We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation. Utilizing the solution of the indefinite moment problem given by M. G. Krein and H. Langer, we show that the conservative Camassa-Holm equation is integrable by the inverse spectral transform in the multi-peakon case.

We discuss direct and inverse spectral theory of self-adjoint Sturm-Liouville relations with separated boundary conditions in the left-definite setting. In particular, we develop singular Weyl-Titchmarsh theory for these relations. Consequently, we apply de Branges' subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to solve the inverse spectral problem associated with the Camassa-Holm equation.2010 Mathematics Subject Classification. Primary 34L05, 34B20; Secondary 46E22, 34B40.[11], [23] for further information. Associated with the Camassa-Holm equation is the left-definite Sturm-Liouville problemon the real line. Direct, and in particular inverse spectral theory of this weighted Sturm-Liouville problem are of peculiar interest for solving the Camassa-Holm equation. Provided ω is strictly positive and smooth enough, it is possible to transform this problem into a Sturm-Liouville problem in potential form and some inverse spectral theory may be drawn from this. However, in order to incorporate the main interesting phenomena (wave breaking [10] and multi-peakon solutions [2], [12]) of the dispersionless Camassa-Holm equation, it is necessary to allow ω at least to be an arbitrary finite signed Borel measure on R. In [5], [7] the authors were able to prove an inverse uniqueness result under some restrictions on the measure ω, which for example prohibits the case of multi-peakon solutions of the Camassa-Holm equation. Using the results of the present paper we are able to avoid these restrictions and to cover the case of arbitrary real finite measures ω; see [17].Note that this application also requires us to consider our Sturm-Liouville problem (1.1) with measure coefficients. For further information on measure Sturm-Liouville equations see e.g. [4] or [18] and the references therein. Moreover, the fact that we allow the weight measure to vanish on arbitrary sets, makes it necessary to work with linear relations instead of operators. Regarding the notion of linear relations, we refer to e.g. [1], [13], [15], [16], [21] or for a brief review, containing all facts which are needed here [18, Appendix B].The paper is organized as follows. After some preliminaries about left-definite measure Sturm-Liouville equations, we give a review of linear relations associated with (1.1) in a modified Sobolev space. In particular, we discuss self-adjoint realizations with separated boundary conditions in Section 3. Since a lot of this first part are minor generalizations of results in e.g. [6], [7], [18], we will omit most of the proofs. In the consecutive two sections we develop Weyl-Titchmarsh theory for such self-adjoint relations. This part can essentially be done along the lines of the singular Weyl-Titchmarsh theory, recently introduced in [20] and [25] for Schrödinger operators. Section 6 introduces some de Branges spaces associated with a left-definite self-adjoint Sturm-Liouville problem. Moreover, we provide crucial properties of these spaces, whi...