Decay of small odd solutions for long range Schr\"odinger and Hartree equations in one dimension

Abstract

We consider the long time asymptotics of (not necessarily small) odd solutions to the nonlinear Schr\"odinger equation with semi-linear and nonlocal Hartree nonlinearities, in one dimension of space. We assume data in the energy space $H^1(\mathbb{R})$ only, and we prove decay to zero in compact regions of space as time tends to infinity. We give three different results where decay holds: semilinear NLS, NLS with a suitable potential, and defocusing Hartree. The proof is based on the use of suitable virial identities, in the spirit of nonlinear Klein-Gordon models as in Kowalczyk-Martel-Mu\~noz, and covers scattering sub, critical and supercritical (long range) nonlinearities. No spectral assumptions on the NLS with potential are needed.