Focusing NLS with inverse square potential

TL;DR

This paper proves radial scattering for the focusing nonlinear Schrödinger equation with inverse square potential in higher dimensions, extending previous results to a broader setting by establishing new dispersive estimates.

Contribution

It extends radial scattering results to higher dimensions for the focusing NLS with inverse square potential, using novel dispersive estimates for radial functions.

Findings

Established radial scattering in dimensions d≥3.

Extended previous results to higher dimensions.

Developed dispersive estimates for radial functions with inverse square potential.

Abstract

In this paper, we utilize the method in Dodson-Murphy [4] to establish the radial scattering result for the focusing nonlinear Schr\"odinger equation with inverse square potential $i\pa_tu-\la u=-|u|^{p-1}u$ in the energy space $H^1_a(\R^d)$ in dimensions $d\geq3$, which extends the result of [10,11] to higher dimensions cases but with radial initial data. The new ingredient is to establish the dispersive estimate for radial function and overcome the weak dispersive estimate when $a<0$.