The Krein-von Neumann extension revisited

TL;DR

This paper provides a detailed analysis of the Krein-von Neumann extension for strictly positive symmetric operators, especially applied to singular Sturm-Liouville operators, with explicit boundary condition formulations.

Contribution

It offers an explicit characterization of the Krein-von Neumann extension for singular Sturm-Liouville operators with three coefficients on arbitrary intervals.

Findings

Explicit boundary conditions for the Krein-von Neumann extension derived

Extension formulas expressed in terms of generalized boundary values

Application to singular Sturm-Liouville operators on arbitrary intervals

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.