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On Spectral Properties of Restricted Fractional Laplacians with Self-adjoint Boundary Conditions on a Finite Interval

Jussi Behrndt, Markus Holzmann, Delio Mugnolo

TL;DR

This work develops a boundary-triplet framework for the restricted fractional Laplacian on a finite interval with $a\in(\tfrac{1}{2},1)$, parametrizing all self-adjoint realizations by 2-by-2 boundary data and establishing discrete, semibounded spectra. It delivers explicit $\gamma$-field and Weyl function formulas, including $\gamma^{(a)}(0)$ and $M^{(a)}(0)$, and uses Krein's formula to obtain resolvent expressions for general extensions, such as Friedrichs (Dirichlet), Krein--von Neumann, and Neumann-type realizations. A key finding is that, unlike the classical Laplacian, the Neumann-type extension for the fractional case has a simple negative eigenvalue, reflecting non-Markovian behavior. The analysis relies on Hörmander transmission spaces to define natural domains and to develop trace theory, illustrating how abstract extension theory yields precise spectral information for nonlocal operators on bounded domains and offering a framework for further spectral comparisons and estimates.

Abstract

We describe all self-adjoint realizations of the restricted fractional Laplacian $(-Δ)^a$ with power $a \in (\frac{1}{2}, 1)$ on a bounded interval by imposing boundary conditions on the functions in the domain of a maximal realization; such conditions relate suitable weighted Dirichlet and Neumann traces. This is done in a systematic way by using the abstract concept of boundary triplets and their Weyl functions from extension and spectral theory of symmetric and self-adjoint operators in Hilbert spaces. Our treatment follows closely the well-known one for classical Laplacians on intervals and it shows that all self-adjoint realizations have purely discrete spectrum and are semibounded from below. To demonstrate the method, we focus on three self-adjoint realizations of the restricted fractional Laplacian: the Friedrichs extension, corresponding to Dirichlet-type boundary conditions, the Krein--von Neumann extension, and a Neumann-type realization. Notably, the Neumann-type realization exhibits a simple negative eigenvalue, thus it is not larger than the Krein--von Neumann extension.

On Spectral Properties of Restricted Fractional Laplacians with Self-adjoint Boundary Conditions on a Finite Interval

TL;DR

This work develops a boundary-triplet framework for the restricted fractional Laplacian on a finite interval with , parametrizing all self-adjoint realizations by 2-by-2 boundary data and establishing discrete, semibounded spectra. It delivers explicit -field and Weyl function formulas, including and , and uses Krein's formula to obtain resolvent expressions for general extensions, such as Friedrichs (Dirichlet), Krein--von Neumann, and Neumann-type realizations. A key finding is that, unlike the classical Laplacian, the Neumann-type extension for the fractional case has a simple negative eigenvalue, reflecting non-Markovian behavior. The analysis relies on Hörmander transmission spaces to define natural domains and to develop trace theory, illustrating how abstract extension theory yields precise spectral information for nonlocal operators on bounded domains and offering a framework for further spectral comparisons and estimates.

Abstract

We describe all self-adjoint realizations of the restricted fractional Laplacian with power on a bounded interval by imposing boundary conditions on the functions in the domain of a maximal realization; such conditions relate suitable weighted Dirichlet and Neumann traces. This is done in a systematic way by using the abstract concept of boundary triplets and their Weyl functions from extension and spectral theory of symmetric and self-adjoint operators in Hilbert spaces. Our treatment follows closely the well-known one for classical Laplacians on intervals and it shows that all self-adjoint realizations have purely discrete spectrum and are semibounded from below. To demonstrate the method, we focus on three self-adjoint realizations of the restricted fractional Laplacian: the Friedrichs extension, corresponding to Dirichlet-type boundary conditions, the Krein--von Neumann extension, and a Neumann-type realization. Notably, the Neumann-type realization exhibits a simple negative eigenvalue, thus it is not larger than the Krein--von Neumann extension.
Paper Structure (14 sections, 22 theorems, 141 equations)

This paper contains 14 sections, 22 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Boundary triplets and Weyl functions
  3. Boundary triplets and self-adjoint extensions
  4. The Weyl function and Krein's formula for self-adjoint extensions
  5. Semibounded self-adjoint extensions
  6. Some properties of Hörmander transmission spaces
  7. Distributions and Sobolev spaces
  8. The Hörmander transmission spaces $H^{\mu(s)}([\alpha, \beta])$
  9. Self-adjoint fractional Laplacians
  10. The minimal, the maximal, and the Dirichlet realization of $(-\Delta)^a$ on $(\alpha, \beta)$
  11. The boundary triplet and self-adjoint extensions
  12. The $\gamma$-field and Weyl function
  13. Further facts about $(-\Delta)^a_{\rm D}$
  14. Semibounded self-adjoint extensions

Key Result

Proposition 2.2

Let $\mu\in\mathbb{C}$, and let $\{{\mathcal{G}},\Upsilon_0,\Upsilon_1\}$ be a boundary triplet for $S^*$. Then $\Theta$ is a linear relation in ${\mathcal{G}}$ with $\mu \in \rho(\Theta)$ if and only if $\Theta$ admits the representation with $\mathcal{A},\mathcal{B}\in\mathcal{L}(\mathcal{G})$ and ${\mathcal{B}} - \mu {\mathcal{A}}$ is bijective. With this choice the adjoint of $\Theta$ is and

Theorems & Definitions (42)

  • Definition 2.1
  • Proposition 2.2
  • Example 2.3
  • Theorem 2.4
  • Definition 2.5
  • Example 2.6
  • Theorem 2.7
  • Proposition 2.8
  • Example 2.9
  • Proposition 2.10
  • ...and 32 more