Sturm–Liouville operators with measure-valued coefficients

Abstract

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 Sturm– Liouville 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. Sturm–Liouville problems (1.1) − d dx p(x) dy dx (x) 1.