Double scattering channels for 1D NLS in the energy space and its generalization to higher dimensions

Abstract

We consider a class of 1D NLS perturbed with a steplike potential. We prove that the nonlinear solutions satisfy the double scattering channels in the energy space. The proof is based on concentration-compactness/rigidity method. We prove moreover that in dimension higher than one, classical scattering holds if the potential is periodic in all but one dimension and is steplike and repulsive in the remaining one.

Dynamics of solutions to the Gross-Pitaevskii equation describing dipolar Bose-Einstein condensates

“…As already mentioned in point (iii-a), the scattering result given in Theorem 1.1 is given by running a concentration/compactness and rigidity scheme, as pioneered by Kenig and Merle in their celebrated works [29,30]. Nowadays there is a huge literature on this method, applied to several dispersive models, and since the scope of this review paper is not to go over the details of these techniques, we refer the reader to [1,12,17,21,24,27] for mass-energy intracritical NLS equations. Let us only mention that the method can be viewed as an induction of the energy method, and it proceeds by contradiction, by assuming the the threshold for global and scattering solutions is strictly smaller than the claimed one.…”

Scattering for NLS with a sum of two repulsive potentials

“…with β > 4 3 . It is the (improved) multi-dimensional analog of the decay assumption arising in [11].…”

An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential

“…In the context of the Schrödinger equation it means that for sufficiently regular and fast decaying right hand sides f the unique solution of the initial value problem i∂ t ψ − ∆ψ + V ψ = f in R n , ψ(0) = ψ 0 , with V as in (1) splits up into two pieces as t → ±∞ that correspond to the two different values of V at infinity. This phenomenon is mathematically understood in the one-dimensional case n = 1 [24,Theorem 1.2], see also [12,13]. One byproduct of our results is that such a splitting into two pieces may as well be observed for the solutions of the corresponding Helmholtz equations in R n which are obtained through the Limiting Absorption Principle, see for instance the formula (16) where the two parts f (x, y)1 (0,∞) (±y) of the right hand side contribute differently to the LAP-solution of the Helmholtz equation.…”