Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model: Spectral Theory and Numerics

Abstract

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 Schr\"odinger operator $\mathbf{M}$. The super-symmetric partner operator is a diagonal matrix whose diagonal entries are strongly singular radial Schr\"odinger 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.