Scattering theory for nls with inverse-square potential in 2d

TL;DR

This paper establishes scattering results for the radial nonlinear Schrödinger equation with an inverse-square potential in two dimensions, extending previous work to a new dimension and employing a modified analytical approach.

Contribution

It extends scattering theory for NLS with inverse-square potential to 2D using a novel adaptation of existing methods.

Findings

Proves scattering below the ground state for 2D radial NLS with inverse-square potential.

Extends previous results from higher dimensions to 2D.

Introduces a modified approach based on Arora-Dodson-Murphy's method.

Abstract

In this paper, we study the long time behavior of the solution of nonlinear Schr\"odinger equation with a singular potential. We prove scattering below the ground state for the radial NLS with inverse-square potential in dimension two $$iu_t+\Delta u-\frac{a u}{|x|^2}= -|u|^pu$$ when $2<p<\infty$ and $a>0$. This work extends the result in [13, 14, 16] to dimension 2D. The key point is a modified version of Arora-Dodson-Murphy's approach [2].