The uniqueness of the ground state and the dynamics of nonlinear Schrödinger equation with inverse square potential

Kai Yang; Chongchun Zeng; Xiaoyi Zhang

doi:10.48550/arxiv.2603.10338

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The uniqueness of the ground state and the dynamics of nonlinear Schrödinger equation with inverse square potential

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The uniqueness of the ground state and the dynamics of nonlinear Schrödinger equation with inverse square potential

Kai Yang, Chongchun Zeng and Xiaoyi Zhang

ArXiv.org

Cornell University

03/11/2026

DOI: 10.48550/arxiv.2603.10338

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Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

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|ᵖufor all0<p<(4/(d-2))in dimensionsd≥ 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 dimensionsd=3, 4, 5 .

Mathematics - Analysis of PDEs

Details

Title: Subtitle: The uniqueness of the ground state and the dynamics of nonlinear Schrödinger equation with inverse square potential
Creators: Kai Yang - Southeast University
Chongchun Zeng - Georgia Institute of Technology
Xiaoyi Zhang - University of Iowa
Resource Type: Preprint
Publication Details: ArXiv.org
DOI: 10.48550/arxiv.2603.10338
ISSN: 2331-8422
Publisher: Cornell University; Ithaca, New York
Language: English
Date posted: 03/11/2026
Academic Unit: Mathematics
Record Identifier: 9985147183502771

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