Spectral properties and bound states of the Dirac equation on periodic quantum graphs

Zhipeng Yang; Ling Zhu

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arXiv · 2601.22603

Spectral properties and bound states of the Dirac equation on periodic quantum graphs

Zhipeng Yang·Ling Zhu

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Abstract

We investigate nonlinear Dirac equations on a periodic quantum graph $G$ and develop a variational approach to the existence and multiplicity of bound states. After introducing the Dirac operator on $G$ with a $\mathbb Z^{d}$-periodic potential, we describe its spectral decomposition and work in the natural energy space. Under asymptotically linear or superquadratic assumptions on the nonlinearity, we establish the required linking geometry and a Cerami-type compactness property modulo $\mathbb Z^{d}$-translations. As a consequence, we prove the existence of at least one bound state and, when the nonlinearity is even, infinitely many geometrically distinct bound states.

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Zhipeng Yang, Ling Zhu. 2026-01-30. Spectral properties and bound states of the Dirac equation on periodic quantum graphs. https://arxiv.org/abs/2601.22603

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