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Journal of Lie Theory 23 (2013), No. 3, 779--794
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


\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 power-like nonlinearity $F$ of degree $\gamma$. We first obtain dispersive estimates and Strichartz estimates with no admissibility conditions. We next deduce global well-posedness 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 well-posedness for arbitrary $L^2$ data. Notice that, in contrast with the Euclidean case, these global well-posedness 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

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