Dispersive estimates for the Schrödinger equation in dimensions four and five

Dispersive Estimates for the Schrödinger Equation for C

We investigate L → L∞ dispersive estimates for the Schrödinger equation iut − ∆u + V u = 0 in odd dimensions greater than three. We obtain dispersive estimates under the optimal smoothness condition for the potential, V ∈ C(n−3)/2(Rn), in dimensions five and seven. We also describe a method to extend this result to arbitrary odd dimensions.

On the stability of ground states in 4D and 5D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equation in four and five space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L) nonlinearities. We show that the center manifold formed by localized in space periodic in time solutions (bound states) is an attractor for all solutions with a...

Dispersive estimates of solutions to the Schrödinger equation in dimensions n≥4

We prove dispersive estimates for solutions to the Schrödinger equation with a real-valued potential V ∈ L∞(R), n ≥ 4, satisfying V (x) = O(〈x〉−(n+2)/2−ǫ), ǫ > 0.

un 2 00 3 Dispersive estimates for Schrödinger operators in dimensions one and three

The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

We discuss the wellposedness theory of the Cauchy problem for the nonlinear Schrödinger equation on compact Riemannian manifolds. New dispersive estimates on the linear Schrödinger group are used to get global existence in the energy space on arbitrary surfaces and three-dimensional manifolds, generalizing earlier results by Bourgain on tori. On the other hand, on specific manifolds such as sph...