Schrödinger equation on Damek-Ricci spaces

Jean-Philippe Anker (1) , Vittoria Pierfelice (1) , Maria Vallarino (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 Dipartimento di Matematica e Applicazioni [Milano]

In this paper we consider the Laplace-Beltrami operator Δ on Damek-Ricci spaces and derive pointwise estimates for the kernel of exp(τΔ), when τ∈C* with Re(τ)≥0. When τ∈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.

Mots clés: Damek-Ricci spaces; Schrödinger equation; heat kernel estimate; dispersive estimate; Strichartz estimate

Domaines: Equations aux dérivées partielles [math.AP]; Analyse classique [math.CA]

Jean-Philippe Anker, Vittoria Pierfelice, Maria Vallarino. Schrödinger equation on Damek-Ricci spaces. Communications in Partial Differential Equations, 2011, 36 (6), pp.976-997. ⟨10.1080/03605302.2010.539658⟩. ⟨hal-00525155⟩