We prove a family of identities that involve the solution u of the Cauchy problem: iâˆ‚tu + âˆ†u = 0, u(0, x) = f, (t, x) âˆˆ Rt Ã—Rx , and the á¸¢ 1 2 (Rn)-norm of the initial datum f . As a consequence ofâ€¦ (More)

We consider a nonlinear SchrÃ¶dinger equation iut âˆ’ h0u + Î²(|u| )u = 0 , (t, x) âˆˆ R Ã— R with h0 = âˆ’ d dx2 + P (x) a SchrÃ¶dinger operator with finitely many spectral bands. We assume the existence ofâ€¦ (More)

where u(t, x) is the unique solution of (1.1), see [12]. Notice that estimates (1.2) describe a certain regularity for the solutions of (1.1) in terms of summability but they do not give any gain ofâ€¦ (More)

We prove Strichartz estimates for the SchrÃ¶dinger equation with an electromagnetic potential, in dimension n â‰¥ 3. The decay and regularity assumptions on the potentials are almost critical, i.e.,â€¦ (More)

We consider the cubic nonlinear SchrÃ¶dinger equation posed on the spatial domain R Ã— Td . We prove modified scattering and construct modified wave operators for small initial and final dataâ€¦ (More)

Following the original approach introduced by T. Cazenave and P.L. Lions in [4] we prove the existence and the orbital stability of standing waves for the following class of NLS: (0.1) iâˆ‚tu + âˆ†u âˆ’ Vâ€¦ (More)

We consider the following family of Cauchy problems: iâˆ‚tu = âˆ†u âˆ’ u|u| , (t, x) âˆˆ R Ã— R u(0) = Ï† âˆˆ H(R) where 0 < Î± < 4 dâˆ’2 for d â‰¥ 3 and 0 < Î± < âˆž for d = 1, 2. We prove that the L-norms of theâ€¦ (More)