The uniqueness of the ground state and the dynamics of nonlinear Schr\"odinger equation with inverse square potential

TL;DR

This paper offers a new proof of the uniqueness of ground state solutions for nonlinear Schrödinger equations with inverse square potential, and analyzes their stability and dynamics across various dimensions.

Contribution

It introduces a classical shooting method approach to establish ground state uniqueness with singular potentials, complementing prior functional analytic methods.

Findings

Uniqueness of ground states for all subcritical powers in dimensions ≥3

Construction of stable and unstable manifolds for ground states

Classification of solutions on the mass-energy level surface

Abstract

In this paper, we first provide an alternative proof of the uniqueness of the ground state solution for NLS with inverse square potential and power nonlinearity $|u|^pu$ for all $0<p<\frac 4{d-2}$ in dimensions $d\ge 3$. While the uniqueness result was previously obtained by Mukherjee-Nam-Nguyen using a functional analytic approach, our method successfully adapts the classical ``shooting method'' to the case with the singular potential, accompanied by a more detailed analysis on the ground state equation. Based upon this result and a comprehensive spectral analysis, we construct the stable/unstable manifolds of the ground state standing wave solutions and classify solutions on the mass-energy level surface of the ground state in dimensions $d=3, 4, 5$.