weyl-titchmarsh theory for sturm-liouville operators with distributional potentials

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 \(p \neq 0\), \(r \gt 0\) 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).