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arXiv · 2508.18021

WKB structure in a scalar model of flat bands

Semyon Dyatlov·Henry Zeng·Maciej Zworski
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Abstract

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.

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Semyon Dyatlov, Henry Zeng, Maciej Zworski. 2025-08-25. WKB structure in a scalar model of flat bands. https://arxiv.org/abs/2508.18021

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