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) ⊆ 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 real-valued and Lebesgue measurable on (a, b), with p 6= 0, r > 0 a.e. on (a, b), and p−1, q, r, s ∈ Lloc((a, b); dx), and f is supposed to satisfy f ∈ ACloc((a, b)), p[f ′ + sf ] ∈ ACloc((a, b)). In particular, this setup implies that τ permits a distributional potential coefficient, including potentials in H−1 loc ((a, b)). We study maximal and minimal Sturm–Liouville operators, all self-adjoint restrictions of the maximal operator Tmax, or equivalently, all self-adjoint extensions of the minimal operator Tmin, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of Tmin. 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 Tmin. Finally, in the special case where τ is regular, we characterize the Krein–von Neumann extension of Tmin and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).