Evolution Equations: Applied and Abstract Perspectives / Equations d'évolution: perspectives appliquées et abstraites

Collection Evolution Equations: Applied and Abstract Perspectives / Equations d'évolution: perspectives appliquées et abstraites

Organizer(s) Disser, Karoline ; Haller-Dintelmann, Robert ; Kyed, Mads ; Saal, Jürgen

Date(s) 28/10/2019 - 01/11/2019

linked URL https://conferences.cirm-math.fr/2071.html

00:00:00 / 00:00:00

4 5

$L^p$-theory for Schrödinger systems

By Abdelaziz Rhandi

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.

Information about the video

Date of recording 28/10/2019
Date of publication 29/11/2019
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19576003
Cite this video Rhandi, Abdelaziz (28/10/2019). $L^p$-theory for Schrödinger systems . CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19576003
URL https://dx.doi.org/10.24350/CIRM.V.19576003

Domain(s)

Analysis of PDEs

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

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