Spectral and Dynamical Analysis of Fractional Discrete Laplacians on the Half-Lattice
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911206280790016 |
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| author | Athmouni, Nassim |
| author_facet | Athmouni, Nassim |
| contents | We investigate discrete fractional Laplacians defined on the half-lattice in several dimensions, allowing possibly different fractional orders along each coordinate direction. By expressing the half-lattice operator as a boundary restriction of the full-lattice one plus a bounded correction that is relatively compact with respect to it, we show that both operators share the same essential spectrum and the same interior threshold structure. For perturbations by a decaying potential, the conjugate-operator method provides a strict Mourre estimate on any compact energy window inside the continuous spectrum, excluding threshold points. As a consequence, a localized Limiting Absorption Principle holds, ensuring the absence of singular continuous spectrum, the finiteness of eigenvalues, and weighted propagation (transport) bounds. The form-theoretic construction also extends naturally to negative fractional orders. Overall, the relative compactness of the boundary correction guarantees that the interior-energy spectral and dynamical results obtained on the full lattice remain valid on the half-lattice without modification. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10680 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral and Dynamical Analysis of Fractional Discrete Laplacians on the Half-Lattice Athmouni, Nassim Spectral Theory Mathematical Physics Analysis of PDEs Functional Analysis 47A10, 47A40, 47B25, 35P25, 81Q10, 39A70 G.2.2; F.1.1 We investigate discrete fractional Laplacians defined on the half-lattice in several dimensions, allowing possibly different fractional orders along each coordinate direction. By expressing the half-lattice operator as a boundary restriction of the full-lattice one plus a bounded correction that is relatively compact with respect to it, we show that both operators share the same essential spectrum and the same interior threshold structure. For perturbations by a decaying potential, the conjugate-operator method provides a strict Mourre estimate on any compact energy window inside the continuous spectrum, excluding threshold points. As a consequence, a localized Limiting Absorption Principle holds, ensuring the absence of singular continuous spectrum, the finiteness of eigenvalues, and weighted propagation (transport) bounds. The form-theoretic construction also extends naturally to negative fractional orders. Overall, the relative compactness of the boundary correction guarantees that the interior-energy spectral and dynamical results obtained on the full lattice remain valid on the half-lattice without modification. |
| title | Spectral and Dynamical Analysis of Fractional Discrete Laplacians on the Half-Lattice |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs Functional Analysis 47A10, 47A40, 47B25, 35P25, 81Q10, 39A70 G.2.2; F.1.1 |
| url | https://arxiv.org/abs/2510.10680 |