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In this article we generalize the Bohr-Sommerfeld rule for scalar symbols ata potential well to matrix-valued symbols having eigenvalues that may coalesceprecisely at the bottom of the well. As an application, we study the existenceof approximately flat bands in moir\'e heterostructures such as strainedtwo-dimensional honeycomb lattices in a model recently introduced by Timmel andMele.

Semiclassical quantization conditions in strained moir\'e lattices