阿贝尔杨-米尔斯-希格斯模型中一阶涡旋的渐近稳定性：线性理论

中文摘要 AI 辅助

我们研究(1+2)维阿贝尔杨-米尔斯-希格斯模型在自对偶耦合下，正交规范中受等变扰动的一阶涡旋附近的线性化动力学。线性化算子是径向L²ᵣₐd(ℝ²;ℝ⁴)上的自伴矩阵薛定谔算子M，其连续谱为[1,∞)，且在唯一的间隙本征值λ²∈(0,1)处存在二维内部模式，这是我们关于基态涡旋渐近稳定性的三篇论文系列的第一部分所确立的。在本系列的第二部分中，我们证明了适用于第三部分的M的线性估计，具体而言，我们证明了色散估计、局部能量衰减估计以及一种转移关系，该关系使我们能够在第三部分的非线性分析中针对平坦克莱因-戈登算子实施时空共振方法。在我们的方法中，证明M的线性估计的核心是与M相关的畸变傅里叶变换，本文前半部分致力于畸变傅里叶变换的构造以及对基础广义本征函数的详细分析，为此我们利用M的超对称分解及其超对称伙伴算子的对角结构，将该问题与两个强奇异标量半直线薛定谔算子的Weyl-Titchmarsh理论关联起来。

英文摘要

We study the linearized dynamics near the degree-one vortex of the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at the self-dual coupling, restricted to equivariant perturbations in the orthogonal gauge. The linearized operator is a selfadjoint matrix Schrödinger operator $\mathbf{M}$ on radial $L^2_{\mathrm{rad}}(\mathbb{R}^2;\mathbb{R}^4)$ with continuous spectrum $[1,\infty)$ and a two-dimensional internal mode at a unique gap eigenvalue $λ^2 \in (0,1)$, as established in Part I of our three-paper series on asymptotic stability of the ground state vortex. In this second part of the series, we prove linear estimates for $\mathbf{M}$ for applications in Part III. Specifically, we prove dispersive and local-energy decay estimates, as well as a transference relation which allows us to implement the space-time resonance method with respect to the flat Klein-Gordon operator in the nonlinear analysis in Part III. The engine for proving linear estimates for $\mathbf{M}$ in our approach is the distorted Fourier transform associated with $\mathbf{M}$. The construction of the distorted Fourier transform together with a detailed analysis of the underlying generalized eigenfunctions occupy the first half of this paper. For this we exploit the super-symmetric factorization of $\mathbf{M}$, and the diagonal structure of the super-symmetric partner operator, to relate the problem to the Weyl-Titchmarsh theory of two strongly singular scalar half-line Schrödinger operators.

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80 pages