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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2511.03532 |
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| _version_ | 1866918187346427904 |
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| author | Wilson, Michael |
| author_facet | Wilson, Michael |
| contents | I study the spectral behavior of the covariant Laplacian $Δ_A = d_A^* d_A$ 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 $Δ_A$. Specifically, if the curvature satisfies the bound $|F_A(x)| \le C(1 + |x|)^{-3-\varepsilon}$ for some $\varepsilon > 0$, then $Δ_A$ is a relatively compact perturbation of the flat Laplacian and hence $σ_{\mathrm{ess}}(Δ_A) = [0,\infty)$. At the critical decay rate $|F_A(x)| \sim |x|^{-3}$, I construct a smooth connection for which $0 \in σ_{\mathrm{ess}}(Δ_A)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03532 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Curvature Decay and the Spectrum of the Non-Abelian Laplacian on $\mathbb{R}^3$ Wilson, Michael Mathematical Physics 35P05 (Primary) 58J50, 81T13 (Secondary) I study the spectral behavior of the covariant Laplacian $Δ_A = d_A^* d_A$ 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 $Δ_A$. Specifically, if the curvature satisfies the bound $|F_A(x)| \le C(1 + |x|)^{-3-\varepsilon}$ for some $\varepsilon > 0$, then $Δ_A$ is a relatively compact perturbation of the flat Laplacian and hence $σ_{\mathrm{ess}}(Δ_A) = [0,\infty)$. At the critical decay rate $|F_A(x)| \sim |x|^{-3}$, I construct a smooth connection for which $0 \in σ_{\mathrm{ess}}(Δ_A)$, 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. |
| title | Curvature Decay and the Spectrum of the Non-Abelian Laplacian on $\mathbb{R}^3$ |
| topic | Mathematical Physics 35P05 (Primary) 58J50, 81T13 (Secondary) |
| url | https://arxiv.org/abs/2511.03532 |