Schrödinger equation on homogeneous trees (Q2856387)

The article is designed around the semilinear Schrödinger equation on a homogeneous tree, associated to Laplace's operator with a power-like nonlinearity of a certain degree. According to Gromov, the homogeneous trees are discrete analogs to the hyperbolic case. The author defines the structure of homogeneous trees and spherical harmonic analysis thereon and, then, formulates the notions of ``dispersive estimate'' and ``Strichartz estimate''. After an analysis of the Schrödinger kernel, he finds the proof for the dispersive estimate and the Strichartz estimate and these estimates are obtained in therms of Bessel functions. Using these two kind of estimates, it is easy to prove the well-posedness and scattering results for the nonlinear Cauchy problem associated to the semilinear Schrödinger equation. According to this, the semilinear Schrödinger equation is globally well-posed for small data, is locally well-posed for arbitrary data and in the case of gauge invariance, local solutions extend to global ones. The author can outline a scattering result in the following form: Under the gauge invariance condition, for any data, the unique global solution to the semilinear Schrödinger equation scatters to linear solutions at infinity.