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H 1 Localization of eigenfunctions via an effective potential

Auteurs : Jerison, David (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : We discuss joint work with Doug Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda. Consider the Neumann boundary value problem for the operator $L = divA\nabla + V$ on a Lipschitz domain $\Omega$ and, more generally, on manifolds with and without boundary. The eigenfunctions of $L$ are often localized, as a result of disorder of the potential $V$, the matrix of coefficients $A$, irregularities of the boundary, or all of the above. In earlier work, Filoche and Mayboroda introduced the function $u$ solving $Lu = 1$, and showed numerically that it strongly reflects this localization. In this talk, we deepen the connection between the eigenfunctions and this landscape function $u$ by proving that its reciprocal $1/u$ acts as an effective potential. The effective potential governs the exponential decay of the eigenfunctions of the system and delivers information on the distribution of eigenvalues near the bottom of the spectrum.

    Codes MSC :
    35P20 - Asymptotic distribution of eigenvalues and eigenfunctions for PD operators
    47A75 - Eigenvalue problems (linear operators)
    81Q10 - Selfadjoint operator theory in quantum theory, including spectral analysis
    81Vxx - Applications of quantum theory to specific physical systems

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 06/10/2017
      Date de captation : 05/10/2017
      Collection : Research talks ; Partial Differential Equations ; Mathematical Physics
      Format : MP4
      Durée : 00:55:29
      Domaine : Mathematical Physics ; PDE
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2017-10-05_Jerison.mp4

    Informations sur la rencontre

    Nom de la rencontre : Harmonic analysis and geometric measure theory / Analyse harmonique et théorie géométrique de la mesure
    Organisateurs de la rencontre : Bernicot, Frédéric ; Durand-Cartagena, Estibalitz ; Lemenant, Antoine ; Pajot, Hervé ; Rigot, Séverine
    Dates : 02/10/2017 - 06/10/2017
    Année de la rencontre : 2017
    URL Congrès : http://conferences.cirm-math.fr/1685.html

    Citation Data

    DOI : 10.24350/CIRM.V.19226303
    Cite this video as: Jerison, David (2017). Localization of eigenfunctions via an effective potential. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19226303
    URI : http://dx.doi.org/10.24350/CIRM.V.19226303


    Voir aussi

    Bibliographie

    1. Arnold, D.N., David, G., Jerison, D., Mayboroda, S., & Filoche, M. (2016). Effective confining potential of quantum states in disordered media. - https://arxiv.org/abs/1505.02684

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