Journal of Lie Theory

no. 3
Schrödinger Equation on Homogeneous Trees
Journal of Lie Theory, Volume 23 (2013) no. 3, pp. 779-794
\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.
DOI: 10.5802/jolt.749
Classification: 35Q55, 43A90, 22E35, 43A85, 81Q05, 81Q35, 35R02
Keywords: Homogeneous tree, nonlinear Schr\"odinger equation, dispersive estimate, Strichartz estimate, scattering 
@article{JOLT_2013_23_3_a9,
     author = {A. J. Eddine},
     title = {Schr\"odinger {Equation} on {Homogeneous} {Trees}},
     journal = {Journal of Lie Theory},
     pages = {779--794},
     year = {2013},
     volume = {23},
     number = {3},
     doi = {10.5802/jolt.749},
     zbl = {1280.35136},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.749/}
}
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%A A. J. Eddine
%T Schrödinger Equation on Homogeneous Trees
%J Journal of Lie Theory
%D 2013
%P 779-794
%V 23
%N 3
%U https://jolt.centre-mersenne.org/articles/10.5802/jolt.749/
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A. J. Eddine. Schrödinger Equation on Homogeneous Trees. Journal of Lie Theory, Volume 23 (2013) no. 3, pp. 779-794. doi: 10.5802/jolt.749

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