Schrödinger equation on Damek-Ricci spaces

Abstract

In this paper we consider the Laplace-Beltrami operator Δon Damek-Ricci spaces and derive pointwise estimates for the kernel of exp(τΔ), when τ\in C* with Re(τ) \geq 0. When τ\in iR*, we obtain in particular pointwise estimates of the Schrödinger kernel associated with Δ. We then prove Strichartz estimates for the Schrödinger 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 Schrödinger 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.