Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials

Abstract

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).