# Scattering in $H^1$ for the intercritical NLS with an inverse-square potential

## TL;DR

This paper investigates the scattering behavior of the intercritical nonlinear Schrödinger equation with an inverse-square potential in dimensions 3 to 6, establishing conditions for scattering and blowup in both focusing and defocusing cases.

## Contribution

It provides the first comprehensive analysis of scattering and blowup phenomena for NLS with inverse-square potential in the intercritical regime across multiple dimensions.

## Key findings

- Scattering established for defocusing NLS with inverse-square potential in $H^1$.
- Dichotomy between scattering and blowup below the ground state in focusing case.
- Results extend understanding of inverse-square potential effects in nonlinear Schrödinger equations.

## Abstract

We study the nonlinear Schr\"odinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.