The Schr\"odinger equation with fractional Laplacian on hyperbolic spaces and homogeneous trees

TL;DR

This paper studies the Schr"odinger equation with fractional Laplacian on hyperbolic spaces and homogeneous trees, revealing differences in dispersive and Strichartz estimates due to geometric properties.

Contribution

It provides new dispersive and Strichartz estimates for fractional Schr"odinger equations on hyperbolic spaces and homogeneous trees, highlighting the impact of geometry on these estimates.

Findings

Loss of derivatives in Strichartz estimates on hyperbolic spaces

No loss of derivatives on homogeneous trees

Dispersive estimates depend on the underlying geometry

Abstract

We investigate dispersive and Strichartz estimates for the Schr\"odinger equation involving the fractional Laplacian in real hyperbolic spaces and their discrete analogues, homogeneous trees. Due to the Knapp phenomenon, the Strichartz estimates on Euclidean spaces for the fractional Laplacian exhibit loss of derivatives. A similar phenomenon appears on real hyperbolic spaces. However, such a loss disappears on homogeneous trees, due to the triviality of the estimates for small times.