Schrödinger equation on Damek-Ricci spaces

Abstract : 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.
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Communications in Partial Differential Equations, Taylor & Francis, 2011, 36 (6), pp.976-997. 〈10.1080/03605302.2010.539658〉
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Jean-Philippe Anker, Vittoria Pierfelice, Maria Vallarino. Schrödinger equation on Damek-Ricci spaces. Communications in Partial Differential Equations, Taylor & Francis, 2011, 36 (6), pp.976-997. 〈10.1080/03605302.2010.539658〉. 〈hal-00525155〉

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