Schrodinger Equation on homogeneous trees

TL;DR

This paper studies the Schrödinger equation on homogeneous trees, deriving dispersive and Strichartz estimates, and establishing global well-posedness and scattering results for small L2 data without gauge invariance constraints.

Contribution

It provides the first dispersive and Strichartz estimates for Schrödinger equations on homogeneous trees, leading to new global well-posedness and scattering results in this setting.

Findings

Dispersive estimates obtained without admissibility conditions.

Global well-posedness for small L2 data without gauge invariance.

Scattering established for small L2 data without gauge invariance.

Abstract

Let T be a homogeneous tree and L the Laplace operator on T. We consider the semilinear Schrodinger equation associated to L with a power-like nonlinearity F of degree d. We first obtain dispersive estimates and Strichartz estimates with no admissibility conditions. We next deduce global well-posedness for small L2 data with no gauge invariance assumption on the nonlinearity F. On the other hand if F is gauge invariant, L2 conservation leads to global well-posedness for arbitrary L2 data. Notice that, in contrast with the Euclidean case, these global well-posedness results hold with no restriction on d > 1. We finally prove scattering for small L2 data, with no gauge invariance assumption.