We make the beginning of the spectral analysis of the anisotropic discrete Maxwell operator $\hat{H}^D$ defined on the square lattice $\mathbb{Z}^3$: we prove that the limiting absorption principle holds. To this aim we construct a conjugate operator to the Fourier series of $\hat{H}^D$ at any not-zero real value. In particular, we analyse the case of thresholds of $\hat{H}^D$.
Nous commençons l’analyse spectrale de l’opérateur de Maxwell discret anisotrope $\hat{H}^D$ défini sur le réseau carré $\mathbb{Z}^3$ : nous prouvons que le principe d’absorption limite est valable. Pour ce faire nous construisons un opérateur conjugué à la série de Fourier de $\hat{H}^D$, en toute valeur réelle non nulle. En particulier, le cas des seuils est résolu.
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Keywords: Maxwell operator, spectral analysis, limiting absorption, threshold, Mourre estimate
Olivier Poisson 1
CC-BY 4.0
@article{AFST_2025_6_34_2_257_0,
author = {Olivier Poisson},
title = {Spectral analysis of the discrete {Maxwell} operator: {The} limiting absorption principle},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {257--314},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 34},
number = {2},
year = {2025},
doi = {10.5802/afst.1813},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1813/}
}
TY - JOUR AU - Olivier Poisson TI - Spectral analysis of the discrete Maxwell operator: The limiting absorption principle JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2025 SP - 257 EP - 314 VL - 34 IS - 2 PB - Université de Toulouse, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1813/ DO - 10.5802/afst.1813 LA - en ID - AFST_2025_6_34_2_257_0 ER -
%0 Journal Article %A Olivier Poisson %T Spectral analysis of the discrete Maxwell operator: The limiting absorption principle %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2025 %P 257-314 %V 34 %N 2 %I Université de Toulouse, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1813/ %R 10.5802/afst.1813 %G en %F AFST_2025_6_34_2_257_0
Olivier Poisson. Spectral analysis of the discrete Maxwell operator: The limiting absorption principle. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 2, pp. 257-314. doi : 10.5802/afst.1813. https://afst.centre-mersenne.org/articles/10.5802/afst.1813/
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