阿贝尔杨-米尔斯-希格斯模型中一阶涡旋的渐近稳定性：谱理论与数值方法
Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model: Spectral Theory and Numerics
AI总结：
本文为三篇系列论文首篇，结合谱理论与数值方法，证明阿贝尔杨-米尔斯-希格斯模型中一阶涡旋在等变扰动下的渐近稳定性，为后续稳定性分析奠定基础。
AI中文摘要：
本文是三篇证明的首篇，研究(1+2)维自对偶耦合下阿贝尔杨-米尔斯-希格斯模型中等变扰动下一阶涡旋的渐近稳定性。在正交规范下，线性化动力学由自伴随矩阵薛定谔算子M控制，其超对称伴算子为对角矩阵，对角元是R²上强奇异径向薛定谔算子。经共轭变换后，谱问题可简化为两个强奇异标量半直线算子的分析。结合分析与严格区间算术，本文证明阈值共振不存在，且M的离散谱恰含一个正间隙本征值（内部模），对应二维本征空间。本文还验证了相关非线性费米黄金规则系数构成确定二次型，可产生内部模的有效非线性阻尼。这些谱分析结果构成后续论文稳定性分析的基础。
英文摘要：
This is the first of three papers proving asymptotic stability of the degree-one vortex under equivariant perturbations in the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at self-dual coupling. In the orthogonal gauge, the linearized dynamics are governed by a selfadjoint matrix Schrödinger operator $\mathbf{M}$. The super-symmetric partner operator is a diagonal matrix whose diagonal entries are strongly singular radial Schrödinger operators on $\mathbb{R}^2$. After a conjugation, this reduces the spectral problem to the analysis of two strongly singular scalar half-line operators. Combining analysis with rigorous interval arithmetic, we prove absence of threshold resonances and show that the discrete spectrum of $\mathbf{M}$ consists of exactly one positive gap eigenvalue (internal mode) with a two-dimensional eigenspace. We also certify that the relevant nonlinear Fermi Golden Rule coefficients form a definite quadratic form, yielding effective nonlinear damping of the internal mode. These spectral inputs form the basis of the stability analysis in the subsequent papers.
