Schr?dinger equation on homogeneous trees

摘要

Let T be a homogeneous tree and L the Laplace operator on T. We consider the semilinear Schr?dinger equation associated to L with a power-like nonlinearity F of degree. 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 γ > 1. We finally prove scattering for arbitrary L~2 data under the gauge invariance assumption.