Schr\"odinger equation on Damek-Ricci spaces

TL;DR

This paper investigates the Schr"odinger equation on Damek-Ricci spaces, deriving kernel estimates, establishing Strichartz estimates, and analyzing dispersive properties, extending known results from hyperbolic spaces to a broader class of spaces.

Contribution

It provides new kernel estimates and Strichartz inequalities for the Schr"odinger equation on Damek-Ricci spaces, expanding the understanding beyond hyperbolic spaces.

Findings

Pointwise estimates for the Schr"odinger kernel on Damek-Ricci spaces

Larger family of admissible pairs for Strichartz estimates

Failure of standard dispersive estimates with a distinguished Laplacian

Abstract

In this paper we consider the Laplace-Beltrami operator \Delta on Damek-Ricci spaces and derive pointwise estimates for the kernel of exp(\tau \Delta), when \tau \in C* with Re(\tau) \geq 0. When \tau \in iR*, we obtain in particular pointwise estimates of the Schr\"odinger kernel associated with \Delta. We then prove Strichartz estimates for the Schr\"odinger equation, for a family of admissible pairs which is larger than in the Euclidean case. This extends the results obtained by Anker and Pierfelice on real hyperbolic spaces. As a further application, we study the dispersive properties of the Schr\"odinger equation associated with a distinguished Laplacian on Damek-Ricci spaces, showing that in this case the standard dispersive estimate fails while suitable weighted Strichartz estimates hold.