Eigenvalue asymptotics and unique continuation of eigenfunctions on planar graphs
[Asymptotique des valeurs propres et prolongement unique des fonctions propres sur un graphe planaire]
Bonnefont, Michel1 ; Golénia, Sylvain1 ; Keller, Matthias2
1 Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400, Talence (France)
2 Universität Potsdam, Institut für Mathematik, 14476 Potsdam (Germany)
Annales de l'Institut Fourier, Online first, 43 p.

Dans ce travail, nous étudions les graphes planaires à courbure négative grande en dehors d’un ensemble fini et plus précisément la théorie spectrale d’opérateurs de Schrödinger sur de tels graphes. Nous obtenons des estimées du premier et du second ordre pour l’asymptotique des valeurs propres. Nous prouvons de plus un résultat de continuation unique ainsi que des propriétés de décroissance des fonctions propres. Les preuves sont basées sur une analyse fine de la géométrie et utilisent une procédure de “copier-coller” qui repose elle-même sur le théorème de Gauß–Bonnet.

We study planar graphs with large negative curvature outside of a finite set and the spectral theory of Schrödinger operators on these graphs. We obtain estimates on the first and second order term of the eigenvalue asymptotics. Moreover, we prove a unique continuation result for eigenfunctions and decay properties of general eigenfunctions. The proofs rely on a detailed analysis of the geometry which employs a Copy-and-Paste procedure based on the Gauß–Bonnet theorem.

Reçu le :
Révisé le :
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DOI : 10.5802/aif.3695
Classification : 47A75, 05C10, 05C63
Keywords: planar graph, Gauß–Bonnet Theorem, Schrödinger operator, eigenvalues asymptotics, unique continuation
Mots-clés : Graphe planaire, Théorème de Gauß–Bonnet, Opérateur de Schrödinger, Asymptotique de valeurs propres, continuation unique

Bonnefont, Michel 1 ; Golénia, Sylvain 1 ; Keller, Matthias 2

1 Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400, Talence (France)
2 Universität Potsdam, Institut für Mathematik, 14476 Potsdam (Germany) 
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Bonnefont, Michel; Golénia, Sylvain; Keller, Matthias. Eigenvalue asymptotics and unique continuation of eigenfunctions on planar graphs. Annales de l'Institut Fourier, Online first, 43 p.

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