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Strict ground-level gaps on long-pendant generalized star graphs for nonlinear Dirac equations

  • Zhipeng Yang
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Abstract

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 dλ∞=dλ,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 ISubset(−mc2,mc2), the presence of a sufficiently long pendant path in the essential reduction of the graph implies dλ(𝒢)<dλ∞. 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 dλ(𝒢)=dλ∞.

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Cite

BibTeXRIS
BibTeX
@misc{yang2026strict,
  title = {{Strict ground-level gaps on long-pendant generalized star graphs for nonlinear Dirac equations}},
  author = {Zhipeng Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24618},
  eprint = {2609.24618},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24618},
}
RIS
TY  - UNPB
TI  - Strict ground-level gaps on long-pendant generalized star graphs for nonlinear Dirac equations
AU  - Zhipeng Yang
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24618
UR  - https://arxiv.org/abs/2609.24618
AB  - 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^∞=d_λ,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 ISubset(−mc^2,mc^2), the presence of a sufficiently long pendant path in the essential reduction of the graph implies d_λ(𝒢)<d_λ^∞. 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(𝒢)=d_λ^∞.
ER  -