Curvature Decay and the Spectrum of the Non-Abelian Laplacian on $\mathbb{R}^3$

TL;DR

This paper characterizes how the decay rate of curvature in non-Abelian gauge connections on b3 determines the essential spectrum of the associated covariant Laplacian, identifying a sharp threshold at decay rate |x|^{-3}.

Contribution

It establishes the precise decay rate of curvature that preserves the essential spectrum of the non-Abelian Laplacian, including explicit constructions at the critical decay.

Findings

Curvature decay faster than |x|^{-3} ensures the essential spectrum remains [0,b3)

At decay rate |x|^{-3}, zero enters the essential spectrum, indicating a sharp threshold

Non-Abelian examples show the commutator term affects spectral behavior at the critical decay

Abstract

I study the spectral behavior of the covariant Laplacian $\Delta_A = d_A^* d_A$ associated with smooth $\mathrm{SU}(2)$ connections on $\mathbb{R}^3$. The main result establishes a sharp threshold for the pointwise decay of curvature governing the essential spectrum of $\Delta_A$. Specifically, if the curvature satisfies the bound $|F_A(x)| \le C(1 + |x|)^{-3-\varepsilon}$ for some $\varepsilon > 0$, then $\Delta_A$ is a relatively compact perturbation of the flat Laplacian and hence $\sigma_{\mathrm{ess}}(\Delta_A) = [0,\infty)$. At the critical decay rate $|F_A(x)| \sim |x|^{-3}$, I construct a smooth connection for which $0 \in \sigma_{\mathrm{ess}}(\Delta_A)$, showing that the threshold is sharp. Moreover, a genuinely non-Abelian example based on the hedgehog ansatz is given to demonstrate that the commutator term $A \wedge A$ contributes at the same order. This work identifies the exact decay rate separating stable preservation of the essential spectrum from the onset of delocalized modes in the non-Abelian setting, providing a counterpart to classical results on magnetic Schr\"odinger operators.