Decay and scattering of solutions to nonlinear Schr\"odinger equations with regular potentials for nonlinearities of sharp growth

TL;DR

This paper establishes decay and scattering results in the energy space for nonlinear Schrödinger equations with regular potentials in higher dimensions, specifically for certain nonlinearities, extending the understanding of long-term behavior of solutions.

Contribution

It proves decay and scattering for nonlinear Schrödinger equations with regular potentials for sharp growth nonlinearities, including the critical case, in dimensions three and higher.

Findings

Decay estimates for solutions in the energy space.

Scattering results for small initial data.

Sharpness of the nonlinearity index for scattering.

Abstract

In this paper, we prove the decay and scattering in the energy space for nonlinear Schr\"odinger equations with regular potentials in $\Bbb R^d$ namely, $i{\partial _t}u + \Delta u - V(x)u + \lambda |u|^{p - 1}u = 0$. We will prove decay estimate and scattering of the solution in the small data case when $1+\frac{2}{d}<p\le1+\frac{4}{d-2}$, $d\ge3$. The index $1+\frac{2}{d}$ is sharp for scattering concerning the result of W. Strauss [21].