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

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].