The Uniqueness of the Ground State and the Dynamics of Nonlinear Schrödinger Equation with Inverse Square Potential

Kai Yang; Chongchun Zeng; Xiaoyi Zhang

doi:10.1137/25M1806636

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The Uniqueness of the Ground State and the Dynamics of Nonlinear Schrödinger Equation with Inverse Square Potential

Journal article

Peer reviewed

The Uniqueness of the Ground State and the Dynamics of Nonlinear Schrödinger Equation with Inverse Square Potential

Kai Yang, Chongchun Zeng and Xiaoyi Zhang

SIAM journal on mathematical analysis, Vol.58(4), pp.4077-4131

08/31/2026

DOI: 10.1137/25M1806636

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Abstract

In this paper, we first provide an alternative proof of the uniqueness of the ground state solution for nonlinear Schrödinger equation (NLS) with inverse square potential and power nonlinearity [Formula: see text] for all [Formula: see text] in dimensions [Formula: see text]. 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 [Formula: see text].

uniqueness

shooting framework

intercritical NLS

stable/unstable manifold

dynamic of solutions

inverse square potential

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: Journal article
Publication Details: SIAM journal on mathematical analysis, Vol.58(4), pp.4077-4131
DOI: 10.1137/25M1806636
ISSN: 0036-1410
eISSN: 1095-7154
Publisher: Society for Industrial and Applied Mathematics
Grant note: Jiangsu Provincial Scientific Research Center of Applied Mathematics: BK20233002 Jiangsu Shuang Chuang Doctoral Plan National Science Foundation: DMS-2350115
The work of the first author was partially supported by the Jiangsu Provincial Sci-entific Research Center of Applied Mathematics grant BK20233002 and the Jiangsu Shuang Chuang Doctoral Plan. The work of the second author was partially supported by the National Science Foundation grant DMS-2350115.
Language: English
Date published: 08/31/2026
Academic Unit: Mathematics
Record Identifier: 9985218380902771

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