Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials (Q2851292)

The authors systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals \((a, b) \subseteq\mathbb R\) associated with rather general differential expressions.NEWLINENEWLINEThe authors study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator, or equivalently, all self-adjoint extensions of the minimal operator, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of the minimal operator. In addition, they 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. They also deal with principal solutions and characterize the Friedrichs extension of the minimal operator.NEWLINENEWLINEFinally, in the special case when the differential expression is regular, the authors characterize the Krein-von Neumann extension of the minimal operator and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents.NEWLINENEWLINEThe results of the paper are new and interesting. The paper gives a contribution to the theory for differential operators.