Curvature Decay and the Spectrum of the Non-Abelian Laplacian on ℝ³

2025/11/05 by Michael Wilson, Wilson, Michael · 1 citation
#math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.2511.03532

Abstract

I study the spectral behavior of the covariant Laplacian Δ_A = d_A^* d_A associated with smooth SU(2) connections on ℝ³. The main result establishes a sharp threshold for the pointwise decay of curvature governing the essential spectrum of Δ_A. Specifically, if the curvature satisfies the bound |F_A(x)| ≤ C(1 + |x|)^-3-ε for some ε > 0, then Δ_A is a relatively compact perturbation of the flat Laplacian and hence σ_ess(Δ_A) = [0,∞). At the critical decay rate |F_A(x)| ∼ |x|^-3, I construct a smooth connection for which 0 ∈ σ_ess(Δ_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 Schrödinger operators.

Citations

• Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds

Cited by

• Threshold Resolvent Singularities and the Infrared Structure of Linearized Gravity

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