Spectral properties and bound states of the Dirac equation on periodic quantum graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910005568995328 |
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| author | Yang, Zhipeng Zhu, Ling |
| author_facet | Yang, Zhipeng Zhu, Ling |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_22603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral properties and bound states of the Dirac equation on periodic quantum graphs Yang, Zhipeng Zhu, Ling Analysis of PDEs 35R02, 35Q41, 81Q35 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. |
| title | Spectral properties and bound states of the Dirac equation on periodic quantum graphs |
| topic | Analysis of PDEs 35R02, 35Q41, 81Q35 |
| url | https://arxiv.org/abs/2601.22603 |