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

Christoph Fischbacher, Jonathan Stanfill

TL;DR

The paper addresses the problem of characterizing nonnegative self-adjoint extensions of singular Sturm--Liouville operators with positive minimal operators, advancing a boundary-values-based parametrization via generalized boundary values and principal/nonprincipal solutions. It develops a comprehensive Weyl--Titchmarsh--Kodaira framework, then explicitly classifies extensions in quasi-regular cases, distinguishing dim2, dim1, and dim0 deficiency scenarios, and shows how fixed-endpoint conditions influence the extension set. A key contribution is the unitary decomposition of reflection-symmetric full-interval problems into direct sums of half-interval problems, enabling simplified spectral analysis for both single and two-interval symmetric configurations. The paper culminates with concrete examples, notably Bessel-type and symmetric Bessel potentials, deriving eigenvalue spectra and applying the results to integral inequalities where the optimal constants arise from half-interval Dirichlet--Neumann eigenvalues, illustrating practical implications in analysis and mathematical physics.

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.

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

TL;DR

The paper addresses the problem of characterizing nonnegative self-adjoint extensions of singular Sturm--Liouville operators with positive minimal operators, advancing a boundary-values-based parametrization via generalized boundary values and principal/nonprincipal solutions. It develops a comprehensive Weyl--Titchmarsh--Kodaira framework, then explicitly classifies extensions in quasi-regular cases, distinguishing dim2, dim1, and dim0 deficiency scenarios, and shows how fixed-endpoint conditions influence the extension set. A key contribution is the unitary decomposition of reflection-symmetric full-interval problems into direct sums of half-interval problems, enabling simplified spectral analysis for both single and two-interval symmetric configurations. The paper culminates with concrete examples, notably Bessel-type and symmetric Bessel potentials, deriving eigenvalue spectra and applying the results to integral inequalities where the optimal constants arise from half-interval Dirichlet--Neumann eigenvalues, illustrating practical implications in analysis and mathematical physics.

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.
Paper Structure (15 sections, 17 theorems, 165 equations)

This paper contains 15 sections, 17 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Singular Weyl--Titchmarsh--Kodaira theory
  3. Nonnegative extensions
  4. General quasi-regular case
  5. The case dim2
  6. The case dim1
  7. The case dim0
  8. Nonnegative extensions for a fixed boundary condition
  9. An application to Sturm--Liouville operators with coefficient functions symmetric about the midpoint of the interval
  10. Two interval problem
  11. Examples and an application to integral inequalities
  12. An application to integral inequalities
  13. Symmetries of the fundamental system of solutions
  14. Factorizations of the characteristic functions
  15. The Krein--von Neumann extension

Key Result

Theorem 2.3

${}$ Assume Hypothesis h2.1. Then the following alternative holds: $(i)$ For every $z\in{\mathbb{C}}$, all solutions $u$ of $(\tau-z)u=0$ are in ${L^2((a,b);rdx)}$ near $a$$($resp., near $b$$)$. $(ii)$ For every $z\in{\mathbb{C}}$, there exists at least one solution $u$ of $(\tau-z)u=0$ which is not

Theorems & Definitions (29)

  • Definition 2.2
  • Theorem 2.3: Weyl's Alternative
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Lemma 2.9
  • Theorem 2.10: GLN20
  • Theorem 2.12: FGKLNS21
  • ...and 19 more