Nonnegative extensions of Sturm-Liouville operators with an application to problems with symmetric coefficient functions

TL;DR

This paper characterizes all nonnegative self-adjoint extensions of singular Sturm-Liouville operators with symmetric coefficients, providing a comprehensive framework and applications including symmetric potentials and integral inequalities.

Contribution

It offers a complete characterization and parameterization of nonnegative self-adjoint extensions for Sturm-Liouville operators with symmetric coefficients, extending to two-interval problems.

Findings

All nonnegative extensions are characterized via boundary values.

Operators with symmetric coefficients are unitarily equivalent to direct sums of half-interval operators.

Application to operators with symmetric Bessel-type potentials and integral inequalities.

Abstract

The purpose of this paper is to study nonnegative self-adjoint extensions associated with singular Sturm-Liouville expressions with strictly positive minimal operators. We provide a full characterization of all possible nonnegative self-adjoint extensions of the minimal operator in terms of generalized boundary values, as well as a parameterization of all nonnegative extensions when fixing a boundary condition at one endpoint. In addition, we investigate problems where the coefficient functions are symmetric about the midpoint of a finite interval, illustrating how every self-adjoint operator of this form is unitarily equivalent to the direct sum of two self-adjoint operators restricted to half of the interval. We also extend these result to symmetric two interval problems. We then apply our previous results to parameterize all nonnegative extensions of operators with symmetric coefficient functions. We end with an example of an operator with a symmetric Bessel-type potential (i.e., symmetric confining potential) and an application to integral inequalities.