A “resonance” here is defined to take place iff W (0) = 0 where W (λ) is the Wronskian of the two Jost solutions at energy λ2, see the following section. It is known that the spectrum of H is purely… (More)

In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimension 1 i ∂ t ψ − △ψ + V ψ = 0, ψ(s) = f (0.1) where V (t, x) is a time-dependent… (More)

The definition of zero energy being a regular point amounts to the same as zero being neither an eigenvalue nor a resonance of H. But the exact meaning of resonance requires some care here, and we… (More)

In this paper we consider various regularity results for discrete quasiperiodic Schrödinger equations −ψn+1 − ψn−1 + V (θ + nω)ψn = Eψn with analytic potential V . We prove that on intervals of… (More)

By this we mean that φ > 0 and φ ∈ C2(R3). It is a classical fact (see Coffman [Cof]) that such solutions exist and are unique for the cubic nonlinearity. Moreover, they are radial and smooth.… (More)

which blows up for t = −ab . Fixing a ∼ 1, b ∼ −1, it is then a natural question to ask whether one may perturb the initial data of (1.2) at time t = 0 such that the corresponding solution exhibits… (More)

The wave equation ∂ttψ −∆ψ − ψ 5 = 0 in R is known to exhibit finite time blowup for large data. It also admits the special static solutions φ(x, a) = (3a) 1 4 (1 + a|x|) 1 2 for all a > 0 which are… (More)

The standing wave solutions of the one-dimensional nonlinear<lb>Schrödinger equation<lb>i∂tψ + ∂ 2<lb>xψ = −|ψ|2σψ<lb>with σ > 2 are well-known to be unstable. In this paper we show that… (More)

By means of the concentration compactness method of Bahouri-Gérard [1] and Kenig-Merle [14], we prove global existence and regularity for wave maps with smooth data and large energy from R → H. The… (More)