Decay and scattering of small solutions of pure power NLS in $\R$ with   $p>3$ and with a potential
  We prove decay and scattering of solutions of the Nonlinear Schr\"oding-er equation (NLS) in ${\mathbf R}$ with pure power nonlinearity with exponent $3<p<5$ when the initial datum is small in $\Sigma$ (bounded energy and variance), in the presence of a linear inhomogeneity represented by a linear potential which is a real valued Schwarz function. We assume absence of discrete modes. The proof is analogous to the one for the translation invariant equation. In particular we find appropriate operators commuting with the linearization. 
Scipio Cuccagna, Vladimir Georgiev, Nicola Visciglia
math.AP math-ph math.MP