WKB structure in a scalar model of flat bands
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914412537839616 |
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| author | Dyatlov, Semyon Zeng, Henry Zworski, Maciej |
| author_facet | Dyatlov, Semyon Zeng, Henry Zworski, Maciej |
| contents | We consider a family of periodic scalar operators for which one can define flat bands in the sense of Floquet-Bloch theory. One puzzling question originating in recent physics literature is a quantisation rule for the values of parameters at which these flat bands occur. We present a general theorem about the structure of solutions to the corresponding equation and a heuristic argument explaining their WKB structure in a specific case. That structure also explains the quantisation condition - both the WKB structure and that rule are confirmed by numerical experiments. Finally, we consider a simplified model in which separation of variables allows the use of complex WKB methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18021 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | WKB structure in a scalar model of flat bands Dyatlov, Semyon Zeng, Henry Zworski, Maciej Mathematical Physics Analysis of PDEs 35H10, 35Q40, 81Q12, 81Q20, 81S10 We consider a family of periodic scalar operators for which one can define flat bands in the sense of Floquet-Bloch theory. One puzzling question originating in recent physics literature is a quantisation rule for the values of parameters at which these flat bands occur. We present a general theorem about the structure of solutions to the corresponding equation and a heuristic argument explaining their WKB structure in a specific case. That structure also explains the quantisation condition - both the WKB structure and that rule are confirmed by numerical experiments. Finally, we consider a simplified model in which separation of variables allows the use of complex WKB methods. |
| title | WKB structure in a scalar model of flat bands |
| topic | Mathematical Physics Analysis of PDEs 35H10, 35Q40, 81Q12, 81Q20, 81S10 |
| url | https://arxiv.org/abs/2508.18021 |