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$L^p$-theory for Schrödinger systems
Multi angle
Authors : Rhandi, Abdelaziz (Author of the conference)
CIRM (Publisher )
35J15 - General theory of second-order, elliptic equations
47D06 - One-parameter semigroups and linear evolution equations
47D08 - Schrödinger and Feynman-Kac semigroups
35J47 - Second-order elliptic systems
CIRM (Publisher )
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Abstract :
In this talk we study for $p\in \left ( 1,\infty \right )$ the $L^{p}$-realization of the vector-valued Schrödinger operator $\mathcal{L}u:= div\left ( Q\triangledown u \right )+Vu$. Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Prüss, and a perturbation theorem by Okazawa, we prove that $L^{p}$, the $L^{p}$-realization of $\mathcal{L}$, defined on the intersection of the natural domains of the differential and multiplication operators which form $\mathcal{L}$, generates a strongly continuous contraction semigroup on $L^{p}\left ( \mathbb{R}^{d} ;\mathbb{C}^{m}\right )$. We also study additional properties of the semigroup such as positivity, ultracontractivity, Gaussian estimates and compactness of the resolvent. We end the talk by giving some generalizations obtained recently and several examples.
Keywords : system of PDE; Schrödinger operator; strongly continuous semigroup
MSC Codes : Keywords : system of PDE; Schrödinger operator; strongly continuous semigroup
35J15 - General theory of second-order, elliptic equations
47D06 - One-parameter semigroups and linear evolution equations
47D08 - Schrödinger and Feynman-Kac semigroups
35J47 - Second-order elliptic systems
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 29/11/2019 Conference Date : 28/10/2019 Subseries : Research talks arXiv category : Analysis of PDEs Mathematical Area(s) : PDE ; Dynamical Systems & ODE Format : MP4 (.mp4) - HD Video Time : 00:28:42 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2019-10-31_Rhandi.mp4 
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Information on the Event
Event Title : Evolution Equations: Applied and Abstract Perspectives / Equations d'évolution: perspectives appliquées et abstraitesEvent Organizers : Disser, Karoline ; Haller-Dintelmann, Robert ; Kyed, Mads ; Saal, Jürgen Dates : 28/10/2019 - 01/11/2019 Event Year : 2019 Event URL : https://conferences.cirm-math.fr/2071.html
Citation Data
DOI : 10.24350/CIRM.V.19576003Cite this video as: Rhandi, Abdelaziz (2019). $L^p$-theory for Schrödinger systems . CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19576003 URI : http://dx.doi.org/10.24350/CIRM.V.19576003 |
See Also
- [Multi angle] Determining the global dynamics of the two-dimensional Navier-Stokes equations by a scalar ODE / Author of the conference Titi, Edriss S..
- [Multi angle] Global well-posedness for the primitive equations coupled to nonlinear moisture dynamics with phase changes / Author of the conference Li, Jinkai.
- [Post-edited] $L^{r}$-Helmholtz-Weyl decomposition in 3D exterior domains / Author of the conference Kozono, Hideo.
- [Multi angle] Dispersive decay for small localized solutions to the Korteweg-de Vries equation / Author of the conference Koch, Herbert.
Bibliography
- KUNZE, Markus, LORENZI, Luca, MAICHINE, Abdallah, et al. ${L^ p} $-theory for Schr\" odinger systems. arXiv preprint arXiv:1705.03333, 2017. - https://arxiv.org/abs/1705.03333
- HIEBER, Matthias, LORENZI, Luca, PRÜSS, Jan, et al. Global properties of generalized Ornstein–Uhlenbeck operators on Lp (RN, RN) with more than linearly growing coefficients. Journal of Mathematical Analysis and Applications, 2009, vol. 350, no 1, p. 100-121. - http://dx.doi.org/10.1016/j.jmaa.2008.09.011
- KUNZE, M., LORENZI, L., MAICHINE, A., et al. Lp-theory for Schrödinger systems, Math. Nachr, vol. 292 n°8 p1763-1776 - https://doi.org/10.1002/mana.201800206
- KUNZE, Markus, MAICHINE, Abdallah, et RHANDI, Abdelaziz. Vector-valued Schr\" odinger operators on $ L^ p $-spaces. arXiv preprint arXiv:1802.09771, 2018. - https://arxiv.org/pdf/1802.09771.pdf
- MAICHINE, Abdallah et RHANDI, Abdelaziz. On a polynomial scalar perturbation of a Schrödinger system in Lp-spaces. Journal of Mathematical Analysis and Applications, 2018, vol. 466, no 1, p. 655-675. - https://arxiv.org/pdf/1802.02772.pdf
