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H 1 $L^p$-theory for Schrödinger systems

Auteurs : Rhandi, Abdelaziz (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : 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

    Codes MSC :
    35J15 - General theory of second-order, elliptic equations
    47D06 - One-parameter semigroups and linear evolution equations
    47D08 - Schrödinger and Feynman-Kac semigroups

      Informations sur la Vidéo

      Réalisateur : Hennenfent, Guillaume
      Langue : Anglais
      Date de publication : 29/11/2019
      Date de captation : 28/10/2019
      Collection : Research talks ; Dynamical Systems and Ordinary Differential Equations ; Partial Differential Equations
      Format : MP4
      Durée : 00:28:42
      Domaine : PDE ; Dynamical Systems & ODE
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2019-10-31_Rhandi.mp4

    Informations sur la rencontre

    Nom de la rencontre : Evolution Equations: Applied and Abstract Perspectives / Equations d'évolution: perspectives appliquées et abstraites
    Organisateurs de la rencontre : Disser, Karoline ; Haller-Dintelmann, Robert ; Kyed, Mads ; Saal, Jürgen
    Dates : 28/10/2019 - 01/11/2019
    Année de la rencontre : 2019
    URL Congrès : https://conferences.cirm-math.fr/2071.html

    Citation Data

    DOI : 10.24350/CIRM.V.19576003
    Cite 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

    Voir aussi


    1. 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

    2. 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

    3. 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

    4. 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

    5. 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