Bands of pure a.c. spectrum for lattice Schr{\"o}dinger operators with a more general long range condition. Part I
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | GOLENIA, Sylvain | |
| hal.structure.identifier | Chercheur indépendant | |
| dc.contributor.author | MANDICH, Marc Adrien | |
| dc.date.accessioned | 2024-04-04T02:47:36Z | |
| dc.date.available | 2024-04-04T02:47:36Z | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/191658 | |
| dc.description.abstractEn | Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov-Vainberg Schr\"odinger operators $H_{\mathrm{std}}= \Delta+V$ and $H_{\mathrm{MV}} = D+V$ on $\ell^2(\mathbb{Z}^d)$, with emphasis on $d=1,2,3$. Considered are electric potentials $V$ satisfying a long range condition of the type: $V-\tau_j ^{\kappa}V$ decays appropriately for some $\kappa \in \mathbb{N}$ and all $1 \leq j \leq d$, where $\tau_j ^{\kappa} V$ is the potential shifted by $\kappa$ units on the $j^{\text{th}}$ coordinate. More comprehensive results are obtained for specific small values of $\kappa$, such as $\kappa =1,2,3,4$. In this article, we work in a simplified framework in which the main takeaway appears to be the existence of bands where a limiting absorption principle holds, and hence absolutely continuous (a.c.) spectrum, for $\kappa>1$ and $\Delta$ (resp.\ $\kappa>2$ and $D$). Other decay conditions for $V$ arise from an isomorphism between $\Delta$ and $D$ in dimension 2. Oscillating potentials are natural examples in application. | |
| dc.language.iso | en | |
| dc.subject.en | 2010 Mathematics Subject Classification. 39A70 | |
| dc.subject.en | 81Q10 | |
| dc.subject.en | 47B25 | |
| dc.subject.en | 47A10 limiting absorption principle | |
| dc.subject.en | discrete Schrödinger operator | |
| dc.subject.en | long range potential | |
| dc.subject.en | Wigner-von Neumann potential | |
| dc.subject.en | 2010 Mathematics Subject Classification. 39A70 | |
| dc.subject.en | Mourre theory | |
| dc.subject.en | weighted Mourre theory | |
| dc.subject.en | Chebyshev polynomials | |
| dc.title.en | Bands of pure a.c. spectrum for lattice Schr{\"o}dinger operators with a more general long range condition. Part I | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Physique mathématique [math-ph] | |
| dc.subject.hal | Mathématiques [math]/Théorie spectrale [math.SP] | |
| dc.subject.hal | Mathématiques [math]/Analyse fonctionnelle [math.FA] | |
| dc.identifier.arxiv | 2102.00726 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| hal.identifier | hal-03118830 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-03118830v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=GOLENIA,%20Sylvain&MANDICH,%20Marc%20Adrien&rft.genre=preprint |
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