发表机构
Yunnan Key Laboratory of Modern Analytical Mathematics and Applications; Department of Mathematics, Yunnan Normal University（云南现代分析数学与应用重点实验室; 云南师范大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究非紧度量图上非线性Dirac方程的严格基态能隙，证明在长悬挂路径存在时能隙严格为正，并刻画了无能隙的纯透射图类。
AI 中文摘要
我们研究非紧度量图上自治非线性Dirac方程的严格基态能隙。相关的无穷远水平是自治全直线水平 \\(d_\lambda^\infty=d_{\lambda,\mathbb R}\\)，因为沿半直线逃逸的集中序列在远离所有顶点重新定心后会看到实直线。我们证明，对于 \\(\lambda\\) 在紧致子区间 \\(I\Subset(-mc^2,mc^2)\\) 内一致地，图的基本约化中存在足够长的悬挂路径意味着 \\[ d_\lambda(\mathcal G)<d_\lambda^\infty. \\] 证明使用了对称全直线基态的半直线限制、一致指数衰减、大规模Dirac-Kirchhoff谱投影的预解局部化，以及广义Nehari纤维的紧致性论证。我们还确定了不存在这种能隙的基本图类：其基本约化与全直线等距的纯透射图满足 \\(d_\lambda(\mathcal G)=d_\lambda^\infty\\)。
英文摘要
We study strict ground-level gaps for autonomous nonlinear Dirac equations on noncompact metric graphs. The relevant level at infinity is the autonomous full-line level \(d_λ^\infty=d_{λ,\mathbb R}\), since a concentrating sequence escaping along a half-line sees the real line after recentering far from all vertices. We prove that, uniformly for \(λ\) in a compact subinterval \(I\Subset(-mc^2,mc^2)\), the presence of a sufficiently long pendant path in the essential reduction of the graph implies \[ d_λ(\mathcal G)<d_λ^\infty. \] The proof uses a half-line restriction of a symmetric full-line ground state, uniform exponential decay, resolvent localization for massive Dirac-Kirchhoff spectral projections, and a compactness argument for generalized Nehari fibres. We also identify elementary graph classes for which no such gap can hold: pure transmission graphs whose essential reduction is isometric to the full line satisfy \(d_λ(\mathcal G)=d_λ^\infty\).
Comments39 pages, comments are welcome
