On Spectral Properties of Restricted Fractional Laplacians with Self-adjoint Boundary Conditions on a Finite Interval
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| Format: | Preprint |
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2025
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| author | Behrndt, Jussi Holzmann, Markus Mugnolo, Delio |
| author_facet | Behrndt, Jussi Holzmann, Markus Mugnolo, Delio |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00094 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Spectral Properties of Restricted Fractional Laplacians with Self-adjoint Boundary Conditions on a Finite Interval Behrndt, Jussi Holzmann, Markus Mugnolo, Delio Spectral Theory Classical Analysis and ODEs Functional Analysis 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. |
| title | On Spectral Properties of Restricted Fractional Laplacians with Self-adjoint Boundary Conditions on a Finite Interval |
| topic | Spectral Theory Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2505.00094 |