
Journal of Lie Theory 23 (2013), No. 3, 779794 Copyright Heldermann Verlag 2013 Schrödinger Equation on Homogeneous Trees Alaa Jamal Eddine MAPMO, Université d'Orléans, Route de Chartres  B.P. 6759, 45067 Orléans 2, France alaa.jamaleddine@univorleans.fr [Abstractpdf] \def\T{{\Bbb T}} Let $\T$ be a homogeneous tree and $\cal L$ the Laplace operator on $\T$. We consider the semilinear Schr\"odinger equation associated to $\cal L$ with a powerlike nonlinearity $F$ of degree $\gamma$. We first obtain dispersive estimates and Strichartz estimates with no admissibility conditions. We next deduce global wellposedness for small $L^2$ data with no gauge invariance assumption on the nonlinearity $F$. On the other hand if $F$ is gauge invariant, $L^2$ conservation leads to global wellposedness for arbitrary $L^2$ data. Notice that, in contrast with the Euclidean case, these global wellposedness results hold for all finite $\gamma\ge 1$. We finally prove scattering for arbitrary $L^2$ data under the gauge invariance assumption. Keywords: Homogeneous tree, nonlinear Schr\"odinger equation, dispersive estimate, Strichartz estimate, scattering. MSC: 35Q55, 43A90; 22E35, 43A85, 81Q05, 81Q35, 35R02 [ Fulltextpdf (183 KB)] for subscribers only. 