带磁场位势的Hartree方程的阈值条件研究

Threshold Conditions for Hartree Equation with Magnetic Potentials

摘要: 本文研究三维空间中带常磁场的Hartree型非线性薛定谔方程的阈值现象。该模型同时包含由磁势引入的外场效应与由Hartree非线性带来的非局部耦合，因此兼具对称性破缺与长程相互作用两方面的分析困难。本文首先建立磁Sobolev空间HA(R3)的基本理论框架，给出能量泛函、守恒律以及 Hartree项所需的关键估计。随后，在适当参数条件下，利用变分方法和集中紧性原理讨论基态解的存在性及其基本性质。本文基于质量保持缩放、Nehari泛函与纤维映射分析，建立适用于带磁场情形的阈值判别框架，说明完整模型在外场作用下不再具有自由Hartree方程那种精确尺度不变性，因此阈值结构应由基态能级与势阱几何来刻画。在此基础上，本文给出解全局存在与有限时间爆破的充分条件，并推导磁场背景下的virial 恒等式。研究表明，磁场虽然显著改变了问题的对称结构和尺度机制，但阈值现象的核心仍然由基态对应的变分几何所支配。

Abstract: This paper studies threshold phenomena of the Hartree-type nonlinear Schrödinger equation with a constant magnetic field in R3. This model includes both external field effects from the magnetic potential and nonlocal coupling due to Hartree nonlinearity. It thus presents analytical difficulties associated with both symmetry breaking and long-range interactions. This paper first establishes the fundamental theoretical framework of magnetic Sobolev spaces and derives the key estimates required for the energy functional, conservation laws, and the Hartree term. Then, under suitable parameter assumptions, we use variational approaches and the concentrationcompactness principle to study the existence and basic properties of ground state solutions. It is shown that the full model no longer possesses the exact scaling invariance of the free Hartree equation under external fields, and hence the threshold structure should be characterized by the ground state energy level and the geometry of the potential well. On this basis, this paper presents sufficient conditions for the global existence and finite-time blowup of solutions, and derives the virial identity in the magnetic field setting. Our study shows that although the magnetic field significantly alters the symmetric structure and scaling mechanism of the problem, the core of the threshold phenomenon is still governed by the variational geometry corresponding to the ground state.