DOI: 10.1515/anona-2025-0201 ISSN: 2191-950X

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

d λ ∞ = d λ , R ${d}_{\lambda }^{\infty }={d}_{\lambda ,\mathbb{R}}$
, 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 ⋐ (− 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 ∞ . $${d}_{\lambda }\left(\mathcal{G}\right){< }{d}_{\lambda }^{\infty }.$$

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 fibers. 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 ∞ ${d}_{\lambda }\left(\mathcal{G}\right)={d}_{\lambda }^{\infty }$
.

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