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arXiv · 2603.10338

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

Kai Yang·Chongchun Zeng·Xiaoyi Zhang
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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 $d=3, 4, 5$.

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Kai Yang, Chongchun Zeng, Xiaoyi Zhang. 2026-03-11. The uniqueness of the ground state and the dynamics of nonlinear Schr\"odinger equation with inverse square potential. https://arxiv.org/abs/2603.10338

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