The Krein–von Neumann extension revisited

Abstract

We revisit the Krein–von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein–von Neumann extension for singular, general (i.e., three-coefficient) Sturm–Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein–von Neumann extension of the strictly positive minimal Sturm–Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.