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Equationsdispersivessurlesvari´et´es Orl´eans,911April2008
On the Schrodinger equation ¨ on Damek–Ricci spaces
Maria Vallarino (Universite´dOrle´ans)
Joint work in progress with Jean-Philippe Anker and Vittoria Pierfelice (Universit´edOrle´ans)
1.Itnroudction:euclideansetting
R n
The NonlinearSchro¨dingerequation iu( t 0 u ( xt )  x =) f +( x Δ) u ( t x ) = F ( u )(1)
has motivated a number of mathematical re-sults: nonlinear optics, Bose-Einstein con-densates,   
The main tool to study (1) is the dispersive estimate for the homogeneous Cauchy prob-lem
t u ( t x ) + Δ u ( t x ) = 0 iu (0  x ) = f ( x )
(2)